Dual-string digital-to-analog converters (DACs), and related circuits, systems, and methods

ABSTRACT

Dual-string digital-to-analog converters (DACs), and related circuits, systems, and methods are disclosed. In embodiments disclosed herein, a primary voltage divider of the dual string-DAC is comprised of at least one adjusting circuit. The adjusting circuit is configured to maintain the ideal voltage of a selected resistor node pair across a secondary voltage divider circuit in response to a primary switch unit selecting a selected resistor node pair. In this manner, impedance isolation is not required between a primary voltage divider and the secondary voltage divider circuit of the dual-string DAC. As a result, as non-limiting examples, the area on an integrated circuit (IC) for a dual-string DAC may be decreased, power consumption of the DAC may be decreased, and/or the dual-string DAC may have increased performance by not requiring a settling time.

RELATED APPLICATION

The present application is related to U.S. patent application Ser. No. 13/834,184, entitled “POLARITY COMPENSATING DUAL-STRING DIGITAL-TO-ANALOG CONVERTERS (DACs), AND RELATED CIRCUITS, SYSTEMS AND METHODS,” filed on Mar. 15, 2013, which is incorporated herein by reference in its entirety.

BACKGROUND

I. Field of the Disclosure

The field of the disclosure relates to dual-string digital-to-analog converters (DACs), and particularly to interconnections and switching of primary and secondary voltage dividers provided therein.

II. Background

A digital-to-analog converter (DAC) is a device that converts digital codes to representative analog signals. For example, the converted analog signals may be recreations of native analog signals previously converted into the digital codes by an analog-to-digital converter (ADC). A common use of ADCs and DACs is converting audio and video signals used in media devices (e.g. televisions, cell phones, MP3 players, etc.) from analog sign representations to digital signal representations, or vice versa.

One type of DAC is a dual-string DAC. A dual resistor string DAC (also referred to as “dual-string DAC”) requires fewer resistors and switches to convert digital codes into analog signal representations as compared to single resistor string DACs. A dual-string DAC includes a first resistor string that generates a coarse conversion of a digital code. A second resistor string of the dual-string DAC generates a finer interpolation of the coarse conversion of the digital code received from the first resistor string to provide an output voltage providing an analog signal representation of the digital code. For example, if a dual-string DAC is configured to convert six (6) bit binary digital codes into sixty-four (64) unique conversions (i.e., 2⁶ conversions), each resistor string of the dual-string DAC could each include eight (8) resistors for a total of sixteen (16) resistors, as opposed to providing sixty-four (64) resistors in a single-string DAC.

For example, FIG. 1 illustrates an exemplary dual-string DAC 10 (referred to herein as “DAC 10”). The DAC 10 functions by applying a received input voltage V_(in) across a primary voltage divider circuit 12, referred to herein as “primary voltage divider 12.” The primary voltage divider 12 provides coarse voltage (i.e., analog signal) values by dividing the input voltage V_(in) across a plurality of primary resistors R(0)-R(N−1) in a primary resistor string 14 at selected resistor node pairs N_(r)(0)-N_(r)(N) at nodes between the primary resistors R(0)-R(N−1). For example, if N equals sixteen (16), this means the number of primary resistors R(0)-R(N−1) provided in the primary voltage divider 12 totals sixteen (16). In this example, the primary voltage divider 12 provides sixteen (16) unique divided primary voltages selectable by four (4) binary bits of a digital code provided to the primary voltage divider 12 for conversion. For example, the bits of a digital DAC input code 15 (hereinafter “DAC input code 15”) are used to select the primary voltages, as illustrated in FIG. 1. In this example, the most significant bits N of the DAC input code 15 are used to select the primary voltages. A coarse divided primary voltage value is selected by a primary switch unit 16 that selects a pair of primary switches U(0)-U(2N−1) to select a selected resistor node pair N, among a plurality of selected resistor node pairs N_(r)(0) to N_(r)(N) in the primary resistor string 14 to select one of the divided primary voltages as a selected coarse divided primary voltage V_(p). This selected coarse divided primary voltage V_(p) is applied across a secondary voltage divider circuit 18, referred to herein as “secondary voltage divider 18.”

With continuing reference to FIG. 1, the secondary voltage divider 18 is provided in the DAC 10 and configured to further divide the selected coarse divided primary voltage V_(p) into a plurality of finer secondary voltages. In this regard, the secondary voltage divider 18 comprises a plurality of secondary resistors R_(s)(0)-R_(s)(Y−1) to form a secondary resistor string 20. Similar to the primary resistor string 14, the secondary resistor string 20 divides the applied primary voltage from the primary voltage divider 12 into finer, interpolated secondary voltages. As the primary voltage is applied across the secondary resistor string 20, a secondary output voltage V_(out) is selected by a secondary voltage divider switch 22. For example, if Y equals thirty-two (32), meaning the number of secondary resistors R_(s)(0)-R_(s)(Y−1) provided in the secondary voltage divider 18 totals thirty-two (32), the secondary voltage divider 18 provides thirty-two (32) unique divided secondary voltages. The thirty-two (32) unique divided secondary voltages are selectable by five (5) binary digital code bits provided to the secondary voltage divider 18. For example, the bits of the DAC input code 15 used to select the secondary voltages may comprise the least significant five (5) bits (LSB) of the DAC input code 15. A finer, interpolated secondary voltage value is selected by the secondary voltage divider switch 22 by selecting a resistor node N_(sr). The selected resistor node N_(sr) is selected from among resistor nodes N_(sr)(0)-N_(sr)(Y) in the secondary resistor string 20 to provide a final, secondary output voltage V_(out) representing the converted DAC input code 15.

When the selected coarse divided primary voltage V_(p) is applied across the secondary resistor string 20 of the secondary voltage divider 18 in the DAC 10 in FIG. 1, the selected primary resistors R(0)-R(N−1) are placed in parallel to the secondary resistor string 20. The parallel placement of the selected primary resistors R(0)-R(N−1) with the secondary resistor string 20 would normally alter the effective resistive characteristics of the selected primary resistors R(0)-R(N−1). The effect of the altered effective resistive characteristics adjusts the selected coarse divided primary voltage V_(p) thereby providing an incorrect selected coarse divided primary voltage V_(p) to the secondary resistor string 20 for the DAC input code 15. To prevent the secondary resistor string 20 from altering the selected coarse divided primary voltage V_(p) across the selected primary resistors R(0)-R(N−1), isolation circuits VF1, VF2 are provided.

With continuing reference to FIG. 1, the isolation circuits VF1, VF2 are disposed between the primary resistor string 14 and the secondary resistor string 20. In this example, the isolation circuits VF1, VF2 are operational amplifiers. The operational amplifiers VF1, VF2 are each configured in a voltage follower mode to maintain the selected coarse divided primary voltage V_(p) applied to the secondary resistor string 20 in this example. The operational amplifiers VF1, VF2 maintain the ideal voltage across the secondary resistor string 20 by isolating the current flow of the primary voltage divider 12 from the secondary voltage divider 18. The effect of isolating the primary voltage divider 12 from the secondary voltage divider 18 is to preserve the native resistive characteristics of the primary voltage divider 12, thus maintaining a predictable, linear voltage division on the primary voltage divider 12 and the secondary voltage divider 18 of the DAC 10. However, providing the operational amplifiers VF1, VF2 comes at the expense of an increase in area usage, consumption of power, and slower performance because the operational amplifiers VF1, VF2 require a settling time.

SUMMARY OF THE DISCLOSURE

Embodiments disclosed in the detailed description include dual-string digital-to-analog converters (DACs), and related circuits, systems, and methods. In embodiments disclosed herein, a primary voltage divider of a dual-string DAC is comprised of at least one adjusting circuit. The adjusting circuit is configured to maintain an ideal voltage of a selected resistor node pair across a secondary voltage divider circuit, in response to a primary switch unit selecting the selected resistor node pair. In this manner, impedance isolation is not required between the primary voltage divider and the secondary voltage divider circuit of the dual-string DAC. As a result, as non-limiting examples, the area on an integrated circuit (IC) for a dual-string DAC may be decreased, power consumption of the dual-string DAC may be decreased, and/or the dual-string DAC may have increased performance by not requiring a settling time.

In this regard, in one embodiment, a primary voltage divider of a dual-string DAC is provided. A primary voltage divider of a dual-string DAC comprises a primary resistor string having a total resistance. The primary resistor string comprises a plurality of resistor nodes configured to divide a DAC input voltage applied across the primary resistor string into a plurality of divided voltage levels. A primary switch unit is configured to receive a DAC input code and select a resistor node circuit among a plurality of resistor node circuits. The resistor node circuit comprises a selected resistor node pair among the plurality of resistor nodes of the primary resistor string based on the DAC input code to couple a divided voltage level across the selected resistor node pair to a secondary voltage divider circuit of the dual-string DAC. The primary voltage divider further comprises at least one adjusting circuit which comprises at least one first fractional resistance to the selected resistor node. The at least one adjusting circuit is configured to maintain an ideal voltage of the selected resistor node pair across the secondary voltage divider circuit in response to the primary switch unit selecting the selected resistor node pair. The ideal voltage is maintained without impedance isolation between the primary voltage divider and the secondary voltage divider circuit. In this manner, as non-limiting examples, area on an integrated circuit (IC) for a DAC may be decreased, power consumption of the DAC may be decreased, and/or the DAC may have increased performance by not requiring a settling time.

In another embodiment, a primary voltage divider of a dual-string DAC for dividing a total voltage across a series of resistive nodes is provided. The primary voltage divider comprises a means for dividing the total voltage across a primary resistor string having a total resistance. The primary resistor string comprises a plurality of resistor nodes configured to divide a DAC input voltage applied across the primary resistor string into a plurality of divided voltage levels. The primary voltage divider further comprises a means for selecting a resistor node circuit comprising a selected resistor node pair among the plurality of resistor nodes of the primary resistor string. The means for selecting the resistor node circuit is based on the DAC input code to couple a divided voltage level across the selected resistor node pair to a secondary voltage divider circuit of the dual-string DAC. The primary voltage divider further comprises a means for adjusting the resistance of the selected resistor node. The means for adjusting the resistance comprises at least one first fractional resistance to maintain an ideal voltage of the selected resistor node pair across the secondary voltage divider circuit in response to the primary switch unit selecting the selected resistor node pair.

In another embodiment, a method of dividing voltage in a dual-string DAC is provided. The method comprises dividing the total voltage and a primary resistor string having a total resistance. The primary resistor string comprises a plurality of resistor nodes configured to divide a DAC input voltage applied across the primary resistor string into a plurality of divided voltage levels. The method further comprises selecting a resistor node circuit comprising a selected resistor node pair among the plurality of resistor nodes of the primary resistor string based on a DAC input code to couple a divided voltage level across the selected resistor node pair to a secondary voltage divider circuit of the dual-string DAC. The method further comprises adjusting the resistance of the selected resistor node with at least one first fractional resistance to maintain an ideal voltage of the selected resistor node pair across the secondary voltage divider circuit in response to a primary switch unit selecting the selected resistor node pair.

BRIEF DESCRIPTION OF THE FIGURES

FIG. 1 is a block diagram of an exemplary dual-string digital-to-analog converter (DAC) that includes a primary voltage divider circuit functioning as a coarse voltage divider, and a secondary voltage divider circuit interpolating coarse voltage selected from the primary voltage divider circuit to generate an analog signal representation of a digital code;

FIG. 2 is an exemplary generalized representation of an adjusting circuit that can be provided in a dual-string DAC, wherein the adjusting circuit is configured to provide fractional resistance to a selected resistor node circuit to maintain an ideal voltage of a selected resistor node pair across a secondary voltage divider circuit;

FIG. 3 is an exemplary dual-string DAC that includes an adjusting circuit comprising a primary resistor and a fractional resistor, wherein the adjusting circuit is configured to maintain an ideal voltage provided by a primary voltage divider circuit across a secondary voltage divider circuit, without requiring impedance isolation between the primary voltage divider circuit and the secondary voltage divider circuit;

FIG. 4 is an exemplary dual-string DAC that includes an adjusting circuit configured to provide fractional resistance to a selected resistor node circuit to maintain an ideal voltage of a selected resistor node pair across a secondary voltage divider circuit;

FIG. 5 is a flowchart of an exemplary process of the dual-string DAC in FIG. 4 converting the DAC input code into a representative analog signal while maintaining the ideal voltage of the selected resistor node pair across the secondary voltage divider circuit;

FIG. 6 is another exemplary dual-string DAC that includes an alternative adjusting circuit of the adjusting circuit in the dual-string DAC of FIG. 3;

FIG. 7 is another exemplary dual-string DAC that includes alternative adjusting circuits to the adjusting circuit in the dual-string DAC of FIG. 3, wherein one of the adjusting circuits is configured to reconfigure coupling of primary resistors between a voltage rail node and the selected resistor node pair, and a ground rail node and the selected resistor node pair, to maintain an ideal voltage of the selected resistor node pair across the secondary voltage divider circuit;

FIG. 8A is an exemplary circuit diagram showing the resistive configuration of the exemplary dual-string DAC of FIG. 6, wherein the primary voltage divider circuit of the dual-string DAC is controlled by a DAC input code of zero (0);

FIG. 8B is an exemplary circuit diagram showing the resistive configuration of the exemplary dual-string DAC of FIG. 6, wherein the primary voltage divider circuit of the dual-string DAC is controlled by a DAC input code of two (2₁₀);

FIG. 9 is an exemplary dual-string DAC including a first and second adjusting circuit configured to provide fractional resistance to a selected resistor node circuit. The first and second adjusting circuits are configured to maintain an ideal voltage of a selected resistor node pair, the first adjusting circuit is coupled between a voltage rail and the selected resistor node pair, and the second adjusting circuit is coupled between a ground rail and the selected resistor node pair;

FIG. 10 is an exemplary dual-string DAC including at least one first adjusting circuit as a part of a selected resistor node pair, and other adjusting circuits coupled between a voltage rail and a ground rail of a primary resistor string, wherein the adjusting circuits in conjunction are configured to maintain an ideal voltage of the selected resistor node pair across a secondary voltage divider circuit;

FIG. 11 is another exemplary dual-string DAC that includes alternative adjusting circuits to the adjusting circuits in the dual-string DAC of FIG. 9, wherein the exemplary dual-string DAC configures the primary switch unit to share coupled fractional resistances;

FIG. 12 is another exemplary dual-string DAC that includes alternative adjusting circuits to the adjusting circuits in the dual-string DAC of FIG. 9, wherein the dual-string DAC configures a primary switch unit to share coupled fractional resistances;

FIG. 13 is another exemplary dual-string DAC that includes alternative adjusting circuits to the adjusting circuits in the dual-string DAC of FIG. 9, wherein the dual-string DAC configures a primary switch unit to share each of at least one first adjusting circuit;

FIG. 14 is another exemplary dual-string DAC that includes alternative adjusting circuits to the adjusting circuits in the dual-string DAC of FIG. 9, wherein the dual-string DAC configures a primary switch unit to share each of at least one first adjusting circuit, wherein an alternative configuration of having a plurality of secondary voltage dividers is shown;

FIG. 15 is another exemplary dual-string DAC including at least one first adjusting circuit configured as a current source coupled to a primary voltage divider circuit, and a second adjusting circuit configured to controllably include at least one second fractional resistance in a total resistance of a primary resistor string, wherein the adjusting circuits in combination are configured to maintain an ideal voltage of a selected resistor node pair across a secondary voltage divider circuit;

FIG. 16 is an exemplary dual-string DAC circuit used to illustrate exemplary polarity and monotonicity issues when the primary switch count provided for each selected resistor node in a primary voltage divider circuit is reduced;

FIG. 17 is an exemplary generalized representation of a secondary voltage divider circuit of a dual-string DAC configured to receive a DAC input code and a polarity indicator, used to control selection of a corresponding secondary switch for selecting a finer, voltage output to be provided as a DAC voltage output of a dual-string DAC;

FIG. 18 is a flowchart of an exemplary generalized process of the dual-string DAC of FIG. 17 for controlling selection of a corresponding secondary switch in a secondary voltage divider circuit for selecting a finer, voltage output to be provided as the DAC voltage output of the dual-string DAC;

FIG. 19 is an exemplary secondary voltage divider circuit of a dual-string DAC, wherein the secondary voltage divider circuit includes a polarity logic switching unit configured to sense a polarity change in a primary voltage divider circuit to maintain monotonicity in the dual-string DAC;

FIG. 20 is an exemplary logic table illustrating a DAC input code and corresponding secondary switch selection in the secondary voltage divider circuit of FIG. 18 to maintain polarity and monotonicity in a dual-string DAC;

FIG. 21 is another exemplary secondary voltage divider circuit of a dual-string DAC, wherein the secondary voltage divider circuit is configured to sense a polarity change in a primary voltage divider circuit output and adjust switch logic using multiplexers and a decoder to maintain monotonicity of the dual-string DAC;

FIG. 22 is an exemplary logic table illustrating a DAC input code and corresponding secondary switch selection in the secondary voltage divider circuit of FIG. 20 to maintain polarity and monotonicity in a dual-string DAC; and

FIG. 23 is a block diagram of an exemplary processor-based system that can include dual-string DACs according to the embodiments disclosed herein, including but not limited to the dual-string DACs of FIGS. 2-22.

DETAILED DESCRIPTION

With reference now to the drawing figures, several exemplary embodiments of the present disclosure are described. The word “exemplary” is used herein to mean “serving as an example, instance, or illustration.” Any embodiment described herein as “exemplary” is not necessarily to be construed as preferred or advantageous over other embodiments.

Embodiments disclosed in the detailed description include dual-string digital-to-analog converters (DACs), and related circuits, systems, and methods. In embodiments disclosed herein, a primary voltage divider of a dual-string DAC is comprised of at least one adjusting circuit. The adjusting circuit is configured to maintain an ideal voltage of a selected resistor node pair across a secondary voltage divider circuit, in response to a primary switch unit selecting the selected resistor node pair. In this manner, impedance isolation is not required between the primary voltage divider and the secondary voltage divider circuit of the dual-string DAC. As a result, as non-limiting examples, the area on an integrated circuit (IC) for a dual-string DAC may be decreased, power consumption of the dual-string DAC may be decreased, and/or the dual-string DAC may have increased performance by not requiring a settling time for the removed impedance isolation circuits.

Other embodiments described below and illustrated by example in FIGS. 15-22, include polarity compensating dual-string digital-to-analog converters (DACs), and related circuits, systems, and methods. In embodiments disclosed herein, a secondary voltage divider of a dual-string DAC includes a switch logic unit. The switch logic unit is configured to compensate for polarity changes in the dual-string DAC to maintain monotonicity in the dual-string DAC. The dual-string DAC being montonic means that the dual-string DAC will convert a digital input code into an representative analog output voltage that increases (or stays constant) or decreases (or stays constant) for a given incremental change in the digital input code. Montonicity may be desired if it is desired for a DAC to convert digital codes to representative analog signals in a linear fashion. The switch logic unit is configured to select a secondary switch among a plurality of secondary switches to divide an input voltage from a primary voltage divider, based on a polarity indicator and the DAC input code, to maintain monotonicity. Each of the secondary switches is connected to a resistor node in a secondary resistor string of the secondary voltage divider. Thus, as a non-limiting example, the dual-string DAC can avoid the need to provide two switches for each resistor node in a primary resistor string to maintain monotonicity.

Before describing embodiments of the polarity compensating dual-string DACs with regard to FIGS. 15-22, examples of dual-string DACs configured to maintain an ideal voltage of a selected resistor node pair across a secondary voltage divider circuit will first be described with regard to FIGS. 2-14.

In this regard, FIG. 2 illustrates an exemplary dual-string DAC 28 configured to maintain an ideal voltage of a selected resistor node pair across a secondary voltage divider circuit. The ideal voltage of the selected resistor node pair across the secondary voltage divider circuit can be maintained without the requirement of providing impedance isolation between a primary voltage divider circuit and a secondary voltage divider circuit. In this regard, the dual-string DAC 28 in this embodiment comprises a primary voltage divider circuit 30 coupled to a secondary voltage divider circuit 32. The primary voltage divider circuit 30 is referred to herein as “primary voltage divider 30.” The secondary voltage divider circuit 32 is referred to herein as “secondary voltage divider 32.” The primary voltage divider 30 divides a DAC input voltage V_(dac) _(—) _(in) 33 to provide a coarse primary output voltage V_(p) to be applied across coarse primary output voltage terminals 34O, 36O, which are coupled to secondary DAC input voltage terminals 34I, 36I, respectively, of the secondary voltage divider 32. The secondary voltage divider 32 divides the coarse primary output voltage V_(p) to provide a DAC output voltage V_(out) 38.

With continuing reference to FIG. 2, the primary voltage divider 30 comprises a primary resistor string 40 and a primary switch unit 42. The DAC input voltage V_(dac) _(—) _(in) 33 is comprised of the voltage between a voltage rail node V_(top) 44 and a ground rail node V_(bot) 46. The DAC input voltage V_(dac) _(—) _(in) 33 is applied across the primary resistor string 40 that is divided in accordance with a most significant bit (MSB) code 48 of the DAC input code 15 in this example. The MSB code 48 comprises a plurality of most significant N binary input bits of the DAC input code 15. In this example, the MSB code 48 controls the primary switch unit 42. The primary switch unit 42 is configured to select a resistor node circuit 47 comprised of a selected resistor node pair N_(r)(i) 49 in this example. The selected resistor node pair N_(r)(i) 49 comprises a selected first resistor node N_(r)(i)H 50 and a selected second resistor node N_(r)(i)L 52. The resistor node N_(r)(i)L 52 will be used to indicate the lower resistor node of a selected resistor node pair N_(r)(i) 49, the voltage at the lower selected resistor node N_(r)(i)L 52 will have a lower relative voltage than the paired higher selected resistor node N_(r)(i)H 50. The selected resistor node pair N_(r)(i) 49 comprises a lower selected resistor node Nr(i)L 52 and a higher selected resistor node N_(r)(i)H 50 based on the MSB code 48. The voltage at the selected resistor node pair N_(r)(i) 49 at the lower selected resistor node N_(r)(i)L 52 and higher selected resistor node N_(r)(i)H 50 is provided to the secondary voltage divider 32 as a secondary DAC input voltage V_(sec) _(—) _(in) across the secondary DAC input voltage terminals 34I, 36I. As discussed below, the secondary voltage divider 32 divides the secondary DAC input voltage V_(sec) _(—) _(in) applied across the secondary DAC input voltage terminals 34I, 36I into a second, finer voltage, which represents the converted DAC input code 15 in analog representation.

With continuing reference to FIG. 2, the divided voltage at the selected resistor node pair N_(r)(i) 49 in an ideal primary resistor string would be a function of the DAC input voltage V_(dac) _(—) _(in) 33 divided by the number of primary resistors of the primary resistor string 40. In such an ideal primary resistor string, without distortions or non-linearities from ancillary or secondary load circuits, such as the secondary voltage divider 32, the voltage at the selected resistor node pair N_(r)(i) 49 will be referred to as an ideal voltage V_(ideal) (not shown). With continuing reference to FIG. 2, if the primary voltage divider 30 and the secondary voltage divider 32 were coupled together without providing an intervening isolating circuit, an actual voltage V_(actual) (not shown) at the selected resistor node pair N_(r)(i) 49 would be different from the ideal voltage V_(ideal), because the secondary load of the secondary voltage divider 32 would be directly coupled to the primary voltage divider 30 without isolation. Thus, the resistive characteristics of the secondary voltage divider 32 would alter or distort the resistance across the selected resistor node pairs N_(r)(i) 49.

With continuing reference to FIG. 2, to adjust for the secondary load created when the primary resistor string 40 is directly coupled to the secondary voltage divider 32 without isolation, the primary resistor string 40 includes a first adjusting circuit 54 in this embodiment. The first adjusting circuit 54 includes a primary resistance R_(p) 56 and at least one first fractional resistance R_(frac1) 58, in this example. In FIGS. 2-14, the first adjusting circuit 54 will comprise the resistor node circuit 47. Thus, as discussed in more detail below, when the primary resistor string 40 is directly coupled to the secondary voltage divider 32 without isolation, the fractional resistance R_(frac1) 58 is coupled to the resistance across the selected resistor node pairs N_(r)(i) 49. The ohmic value of the fractional resistance R_(frac1) 58 is chosen to compensate and provide resistance across the selected resistor node pair N_(r)(i) 49 as if the secondary voltage divider 32 were isolated or not coupled to the primary voltage divider 30. Thus, the ideal voltage V_(ideal) provided by the primary voltage divider 30 to the secondary voltage divider 32, according to the selected resistor node pair N_(r)(i) 49, is maintained. Thus, with the first adjusting circuit 54 in FIG. 2, the need to provide isolation circuits between the primary voltage divider 30 and the secondary voltage divider 32 is not required to maintain the ideal voltage V_(ideal). This is discussed in more detail below starting at FIG. 4. The further dividing of the selected coarse divided primary voltage V_(p) into the DAC output voltage V_(out) 38 will now be discussed with continued reference to FIG. 2.

With continuing reference to FIG. 2, the coarse primary output voltage V_(p) across coarse primary output voltage terminals 34O, 36O is applied across the secondary voltage divider 32. The secondary voltage divider 32 comprises a secondary resistor string 60 and a secondary switch unit 64, otherwise known as a “secondary voltage divider switch 64.” The secondary voltage divider 32 is configured to receive the coarse primary output voltage V_(p) at the coarse primary output voltage terminals 34O, 36O as a secondary DAC input voltage V_(sec) _(—) _(in) applied across the secondary DAC input voltage terminals 34I, 36I. The secondary voltage divider 32 is further configured to receive an LSB code 66. The DAC output voltage V_(out) 38 is selected based on a least significant bit (LSB) code 66 of the DAC input code 15 in this example. The LSB code 66 is also known as the “secondary DAC input code 66.” The LSB code 66 controls the secondary switch unit 64, which is configured to select a DAC output voltage V_(out) 38 from a selected secondary resistor node Nsr(0)-N_(sr)(Y−1), where Y=LSB code 66.

Maintaining an ideal voltage V_(ideal) as applied across the secondary voltage divider 32 of the dual-string DAC 28 in FIG. 2 has three (3) exemplary interconnection principles when the primary voltage divider 30 and the secondary voltage divider 32 are interconnected without impedance isolation. In this regard, FIG. 3 is an exemplary illustration of these three (3) interconnection principles represented by three exemplary interconnection relationships with further reference to FIGS. 1 and 2. The first interconnection relationship 68 illustrates resistance provided in the first adjusting circuit 54 that is coupled to the selected resistor node pair N_(r)(i) 49 in the primary voltage divider 30 to maintain the resistance characteristics of the selected resistor node pair N_(r)(i) 49 as if the secondary voltage divider 32 was impedance isolated from the primary voltage divider 30. In this regard, the first interconnection relationship 68 provides as follows: R _(p) =R _(sd)∥(R _(p) +R _(frac)),

which can be mathematically represented as: 1/(1/(R _(p) +R _(frac))+(1/R _(sd)))=R _(p)

The primary resistance R_(p) 56 and the first fractional resistance R_(frac) 58 of the first adjusting circuit 54 are coupled between the selected lower selected resistor node N_(r)(i)L 52 and the higher selected resistor node N_(r)(i)H 50 of the primary voltage divider 30. The combined primary resistance of the first adjusting circuit 54 comprises the serial resistance of the primary resistance R_(p) 56 and the fractional resistance R_(frac) 58. A combined secondary serial resistance R_(sd) 70 of the secondary resistor string 60 comprises the total serial resistance of the plurality of secondary resistors R_(s)(0)-R_(s)(Y−1) of the secondary resistor string 60. Thus, when the first adjusting circuit 54 is coupled in parallel to the secondary resistor string 60 without impedance isolation, the resistance of the first adjusting circuit 54 (i.e., primary resistance R_(p) 56+fractional resistance R_(frac) 58) is coupled in parallel with the combined secondary serial resistance R_(sd) 70. Thus, the first interconnection relationship 68 is stated as the primary resistance R_(p)=R_(sd)∥(R_(p)+R_(frac)) or 1/(1/(R_(p)+R_(frac)))+1/R_(sd)))=R_(p).

The resistances of the primary resistance R_(p) 56 and the first fractional resistance R_(frac) 58 of the first adjusting circuit 54 are specific to the resistance of the secondary resistor string 60. The resistances of the primary resistance R_(p) 56 and the first fractional resistance R_(frac) 58 are selected such that when the secondary resistor string 60 is coupled to the selected resistor node pairs N_(r)(i) 49, the resistance across the selected resistor node pairs N_(r)(i) 49 is the same as if the secondary voltage divider 32 was impedance isolated from the primary voltage divider 30.

To maintain the ideal voltage V_(ideal) at selected resistor node pair N_(r)(i) 49 in the dual-string DAC 28 in FIG. 2, in addition to maintaining the resistance across the selected resistor node pair N_(r)(i) 49, it is also necessary in this example to maintain the resistance above the selected resistor node pair N_(r)(i) 49 to the voltage rail node V_(top) 44 and the resistance below the selected resistor node pair N_(r)(i) 49 to the ground rail node V_(bot) 46. In this manner, the total resistance of the primary resistor string 40 is adjusted to maintain the ideal voltage V_(ideal) at resistor node pair N_(r)(i) 49, without the need for impedance isolation between the primary resistor string 40 and the secondary resistor string 60. Otherwise, the coarse divided primary output V_(p) divided across the selected resistor node pair N_(r)(i) 49 will be different from its ideal voltage V_(ideal).

With continuing reference to FIG. 3, a second interconnection relationship 72 is provided to illustrate the total resistance value coupled between the voltage rail node V_(top) 44 (FIG. 2) and the higher selected resistor node N_(r)(i)H 50. The resistance between the voltage rail node V_(top) 44 and the higher selected resistor node N_(r)(i)H 50 is adjusted to compensate for adjustments in the selection of the selected resistor node pair N_(r)(i) 49. The second interconnection relationship 72 provides the total resistance that would be coupled between the voltage rail node V_(top) 44 and the higher selected resistor node N_(r)(i)H 50, to maintain the resistance of the primary resistor string 40. In this manner, the voltage at the higher selected resistor node N_(r)(i)H 50 will be maintained equal or substantially equal to the voltage of the equivalent selected resistor node in an ideal primary voltage divider (not shown) with impedance isolation. In this regard, the second interconnection relationship 72 is provided, as follows: (N-i-1)*R _(p) +R _(bulk2),

where R_(bulk2) may be equal to zero (0),

N is the number of selectable selected resistor node pairs N_(r)(i) 49 (i.e., selectable resistor node pairs) in the primary resistor string 40, and

i is the current decoded MSB code 48.

The second interconnection relationship 72 determines the total resistance between the voltage rail node V_(top) 44 and the higher selected resistor node N_(r)(i)H 50. For an ideal primary resistor string 40, the total resistance would be equal or substantially equal to the number of selectable unique resistor node pairs between the voltage rail node V_(top) 44 and the higher selected resistor node N_(r)(i)H 50 multiplied by the primary resistance R_(p) 56. An optional second bulk resistance R_(bulk2) may be included if further calibrations to the resistance are needed based on any biasing in the dual-string DAC 28.

With continuing reference to FIG. 3, a third interconnection relationship 74 is provided to illustrate the total resistance value coupled between the ground rail node V_(bot) 46 (FIG. 2) and the lower selected resistor node N_(r)(i)L 52. The resistance between the ground rail node V_(bit) 46 and the lower selected resistor node N_(r)(i)L 52 is adjusted to compensate for adjustments in the selection of the selected resistor node pair N_(r)(i) 49. The third interconnection relationship 74 provides the total resistance that would be coupled between the ground rail node V_(bot) 46 and the lower selected resistor node N_(r)(i)L 52, to maintain the resistance of the primary resistor string 40. In this manner, the voltage at the lower selected resistor node N_(r)(i)L 52 will be maintained at equal or substantially equal to the voltage of the equivalent selected resistor node in an ideal primary voltage divider (not shown) with impedance isolation. In this regard, the third interconnection relationship 74 is provided, as follows: i*R _(p) +R _(bulk1), where

R_(bulk1) may be equal to zero (0),

N is the number of selectable selected resistor node pairs N_(r)(i) 49 (i.e., selectable resistor node pairs) in the primary resistor string 40, and

i is the current decoded MSB code 48.

The third interconnection relationship 74 determines the total resistance between the ground rail node V_(bot) 46 and the lower selected resistor node N_(r)(i)L 52. For an ideal primary resistor string 40, the total resistance would be equal or substantially equal to the number of selectable unique resistor node pairs between the ground rail node V_(bot) 46 and the lower selected resistor node N_(r)(i)L 52 multiplied by the primary resistance R_(p) 56. An optional second bulk resistance R_(bulk1) may be included if further calibrations to the resistance are needed based on any biasing in the dual-string DAC 28. The ideal voltage will be maintained at Nr(i)L 52 and Nr(i)H 50 when all three interconnection relationships are simultaneously met.

In each of the following exemplary embodiments, the resistances, referred to as the primary resistance R_(p) 56 and the at least one first fractional resistance R_(frac) 58, may be comprised of a single resistor or a plurality of resistor units R_(u). Resistor unit R_(u) is a common resistive unit value that may be combined to total the necessary resistive values for the primary resistance R_(p) 56 and the at least one first fractional resistance R_(frac) 58. It should be noted that based on the design choices, the resistance of resistor unit R_(u) may be a common factor or a common unit included in the primary resistance R_(p) 56 and the first fractional resistance R_(frac) 58.

FIG. 4 is an exemplary embodiment of a dual-string DAC 28(1) configured to maintain an ideal voltage of a selected resistor node pair across a secondary voltage divider 32(1). The dual-string DAC 28(1) in this example comprises a primary resistor string 40(1) coupled to a secondary resistor string 60(1). The primary switch unit 42(1) is configured to select a resistor node circuit 47(1) among a plurality of resistor node circuits, the resistor node circuit comprises a selected resistor node pair N_(r)(i) 49(1). The primary resistor string 40(1) being coupled to the secondary resistor string 60(1) creates a parallel resistance when interconnected without isolation circuits. The parallel resistance created is compensated for in accordance with the first interconnection relationship 68, the second interconnection relationship 72, and the third interconnection relationship 74, as discussed above in FIG. 3. To conform to the first, second, and third interconnection relationships 68, 72, 74 in FIG. 3, the primary resistor string 40(1) is comprised of a plurality of first adjusting circuits 54(1)(0)-54(1)(N−1). The number of first adjusting circuits 54(1) is equal to N, where N is the number (2^(MSB)) of selectable unique resistor node pairs N_(r)(1)(0) to N_(r)(i)(N−1) 49(1) in this example. For purposes of this embodiment, the index “i” will be used to indicate the index of the selected resistor node pair, and “not i” will be used to indicate the index(es) of any non-selected resistor node pairs. For example, if i=three (3) out of a range of 0 to 7, “N_(r)(1)(3) 49(1)” indicates the fourth selected resistor node pair N_(r)(1)(3) 49(1), where the indexes begin at 0. As an example of “not i”, this will indicate any other selected resistor node pair N_(r)(1)(not 3) not represented by the selected resistor node pair N_(r)(1)(3) 49(1).

With continued reference to FIG. 4, a selected unique resistor node pair N_(r)(1)(i) 49(1) from among the selectable unique resistor node pairs N_(r)(1)(0)-N_(r)(1)(N−1) 49(1) is comprised of one of the first adjusting circuits 54(1)(0)-54(1)(N−1). The selected unique resistor node pair N_(r)(1)(i) 49 comprises a primary resistance R_(p) 56(1), at least one first fractional resistance R_(frac1) 58(1), and a first switch S_(p) 1 76(1). One of the at least one first adjusting circuits 54(1)(0)-54(1)(N−1) is configured with the primary resistance R_(p) 56(1) and the at least one first fractional resistance R_(frac1) 58(1) coupled in series. To satisfy the first interconnection relationship 68 relationship in FIG. 3, the first adjusting circuits 54(1)(0)-54(1)(N−1) are further configured with respective first switches S_(p) 1 76(1)(0)-76(1)(N−1) coupled in parallel with corresponding first fractional resistances R_(frac1) 58(1)(0)-58(1)(N−1). The selected resistor node pair N_(r)(1)(i) 49(1) is configured to place the first switch S_(p) 1 76(1)(i) in a coupling mode by opening the first switch S_(p) 1 76(1)(i). The first switch S_(p) 1 76(1)(i) is associated with the selected resistor node pair N_(r)(1)(i) 49(1), thus creating a combined serial resistance of the primary resistance R_(p) 56(1)(i) and the first fractional resistance R_(frac1) 58(1)(i) for the selected resistor node pair N_(r)(1)(i) 49(1). This combined serial resistance, coupled in parallel with the total secondary serial resistance R_(sd) 70 of the secondary resistor string 60(1), will create an effective parallel resistance of R_(p) 56(1)=R_(sd)∥(R_(p)+R_(frac)). The value of the primary resistance R_(p) 56(1) will be determined based in accordance with the first interconnection relationship 68 of R_(p)=R_(sd)∥(R_(p)+R_(frac)) in FIG. 3. The primary resistance Rp 56(1) is a calculated resistance value. Within the selected resistor node pair N_(r)(1)(i) 49(1) the primary resistance Rp 56(1) will be substantially the same. While the calculated value of the primary resistance Rp 56(1) will be substantially the same, it is possible the actual value of the physically coupled resistances between the selected resistor node pair N_(r)(1)(i) 49(1) may vary as required based on design choices made.

With continuing reference to FIG. 4, to satisfy the second interconnection relationship 72 and the third interconnection relationship 74 in FIG. 3, any non-selected resistor node pairs N_(r)(1)(not i) are configured to place the first switches S_(p1) 76(1)(not i) in a decoupling mode. By placing the non-selected resistor node pairs N_(r)(1)(not i) in decoupling mode, the corresponding resistance of the non-selected resistor node pairs N_(r)(1)(not i) will equal or substantially equal the primary resistance R_(p) 56(1). Decoupling mode is where first switches S_(p) 1 76(1)(not i) are closed. By closing the first switch S_(p) 1 76(1)(not i) of the non-selected resistor node pairs N_(r)(1)(not i), a short circuit bypassing the corresponding first fractional resistance R_(frac1) 58(i)(not i) is created. By bypassing the corresponding first fractional resistance R_(frac1) 58(1)(not i) the first adjusting circuits 54(1)(not i) are adjusted to substantially equal the primary resistance R_(p) 56(1). The second interconnection relationship 72 and the third interconnection relationship 74 are met since the first fractional resistance R_(frac1) 58(1)(not i) is removed from the primary resistor string 40(1) when the respective non-selected resistor node pairs N_(r)(1)(not i) are placed in a decoupled mode.

For example, with continuing reference to FIG. 4, as a non-limiting example, MSB code 48(1) has three (3) bits with a bit value of “100₂” equal to decimal four 4₁₀. In normalized resistive units R_(u), primary resistance R_(p) is equal to 4R_(u), fractional resistance R_(frac) 58(1)(4) is equal to 0.5R_(u), and the secondary serial resistance R_(sd) 70(1) is equal to 36R_(u). With these design choices, all three of the first, second, and third interconnection relationships 68, 72, 74 in FIG. 3 are satisfied in this example. The first interconnection relationship 68, R_(p)=R_(sd)∥(R_(p)+R_(frac)), is satisfied based on the equation 1/(1/(R_(p)+R_(frac))+(1/R_(sd)))=R_(p) with the exemplary values above provided in the parallel resistive equation 1/(1/(4R_(u)+0.5R_(u))+(1/36R_(u)))=4R_(u)=1R_(p). In addition, both the second interconnection relationship 72 and the third interconnection relationship 74 in FIG. 3 are satisfied, since the non-selected resistor node pairs N_(r)(1)(not i) are configured to close the first fractional resistance switches S_(p1) 76(1)(not i) associated with the non-selected resistor node pairs N_(r)(1)(not i). The second interconnection relationship 72, (N-i-1)*R_(p)+R_(bulk2), is satisfied, because the total resistance between the voltage rail node V_(top) 44(1) and the selected resistor node pair N_(r)(1)(i) 49(1) equals (N−i−1)*R_(p)+R_(bulk2) 72, where R_(bulk2) equals zero (0). In this example, N is equal to the maximum value of the MSB code 48(1) (e.g., N is equal to eight (8), if MSB code 48(1) has three (3) bits). Also in this example, the selected resistor node pair N_(r)(1)(i) 49(1) is determined by “i,” where “i” is equal to the binary number “100₂” or “4₁₀”, primary resistance R_(p) 56(1)(4)=4R_(u) and R_(bulk2)=zero (0). Based on the second interconnection relationship 72 (N-i-1)*R_(p)+R_(bulk2) from FIG. 3, the resistance between the voltage rail node V_(top) 44(1) and the selected resistor node pair N_(r)(1)(4) 49(1) equals (8-4-1)*4Ru+0. The second interconnection relationship 72 in this example further reduces to 3*4Ru+0 or 12Ru. The second interconnection relationship 72 relationship, (N-i-1)*R_(p)+R_(bulk2) 72, is satisfied since the primary resistance R_(p) 56(1) equals 4Ru. There are three (3) primary resistances R_(p) 56(1) provided between the voltage rail node V_(top) 44(1) and the selected resistor node pair N_(r)(1)(4) 49(1).

With continued reference to FIG. 4, the third interconnection relationship 74, i*R_(p)+R_(bulk1) 74, will also be satisfied using the same example, where R_(bulk1) is equal to zero (0). The equation reduces to 4*(R_(p))+0 or 4R_(p)=16R_(u). In the example in FIG. 4, the first fractional resistance switches S_(p1) 76(i)(not i) are all closed. This will short circuit the corresponding fractional resistance R_(frac) 58(1) for each of the four (4) non-selected resistor node pairs N_(r)(1)(0)-N_(r)(1)(3) 49(1) located between the ground rail node V_(bot) 46(1) and the selected resistor node pair N_(r)(1)(4) 49(1). The total resistance between the ground rail node V_(bot) 46(1) and the selected resistor node pair N_(r)(1)(4) 49(1) is 4R_(p)=16R_(u).

As described above, FIG. 4 illustrates an exemplary embodiment of a dual-string DAC 28(1) that conforms to each of the three first, second, and third interconnection relationships 68, 72, 74 in FIG. 3. However, there are numerous exemplary embodiments that may utilize these first, second, and third interconnection relationships 68, 72, 74 in many possible configurations. These exemplary embodiments can reduce the number of resistors in a dual-string DAC based on the use or application requirements. It is equally important to minimize unused components because valuable space is wasted if a component has to be built into device and at times during operation when the component is not used. Additionally, one design goal of certain embodiments herein is to eliminate, where possible, isolation circuits that can consume large segments of device area and even slow performance as discussed above.

To supplement the discussion of the dual-string DAC 28 above with regard to FIG. 2, FIG. 5 is provided. FIG. 5 is an exemplary process to explain the operation of the dual-string DAC 28. First, the primary voltage divider 30 divides a DAC input voltage (V_(dac) _(—) _(in)) across the primary resistor string 40 having a total resistance into a plurality of coarse divided primary voltages based on the DAC input code 15 (block 78). As discussed above, the primary resistor string 40 comprises a plurality of selectable resistor node pairs N_(r)(i) 49 configured to divide the DAC input voltage (V_(dac) _(—) _(in)) applied across the primary resistor string 40 into the plurality of coarse divided primary voltages. The primary switch unit 42 receives the MSB code 48 of the DAC input code 15 (block 80), the MSB code 48 is decoded and then converted to select the resistor node pair N_(r)(i) 49 (block 81) from among a plurality of resistor node pairs N_(r)(i) 49. The primary switch unit 42 is configured to select a resistor node circuit 47 among a plurality of resistor node circuits, the resistor node circuit comprises a selected resistor node pair N_(r)(i) 49.

In this regard, FIG. 6 illustrates another exemplary embodiment of a dual-string DAC 28(2). This exemplary embodiment reduces the total number of resistances coupled in the primary resistor string 40(2), and eliminates the first switch S_(p) 1 76(1) as discussed in FIG. 4 above. The dual-string DAC 28(2) comprises a primary switch unit 42(2) configured to select resistor node circuit 47(2) among a plurality of resistor node circuits, the resistor node circuit comprises a selected resistor node pair N_(r)(2)(i) 49(2) with a resistance equal to R_(p)(2)+R_(frac)(2). The selection is made wherein the R_(p)(2) and R_(frac)(2) resistance values are a design choice with R_(p) and R_(frac) determined by the first interconnection relationship 68 R_(p)(2)=R_(sd)(2)∥(R_(p)(2)+R_(frac)(2)). The primary switch unit 42(2) is configured to combine a plurality of adjacent resistances such that the combination of adjacent resistances is determined by the three interconnection relationships.

With continued reference to FIG. 6, the primary resistor string 40(2) comprises at least one first adjusting circuit 54(2)(0) to 54(2)(N−1). The first adjusting circuit 54(2)(N−1) is coupled immediately adjacent to the voltage rail node V_(top) 44(2) and is comprised of two resistances, a primary resistance R_(p) 56(2)(N−1) and a first fractional resistance R_(frac1) 58(2)(N−1). Another first adjusting circuits 54(2)(0) is coupled immediately adjacent to the ground rail node V_(bot) 46(2) and is comprised of two resistances a primary resistance R_(p) 56(2)(0) and a first fractional resistance R_(frac1) 58(2)(0). Each of the plurality of first adjusting circuits 54(2)(0) 54(2)(N−1) in this example share the at least one first fractional resistance R_(frac1) 58(2)(1) 58(2)(N−1) with the first adjusting circuits 54(2)(1) and 54(2)(N−2) that are immediately adjacent. However, each of the plurality of first adjusting circuits 54(2)(1) to 54(2)(N−2) use an alternate configuration which still conforms to the first interconnection relationship 68, R_(p)=R_(sd)∥(R_(p)+R_(frac)) 68.

In this regard, in FIG. 6, in each of the plurality of first adjusting circuits 54(2)(1) to 54(2)(N−2) a combination of fractional resistances and primary resistances will be used to construct the total resistance required. The first adjusting circuits 54(2)(1) to 54(2)(N−2) comprise a plurality of primary resistances R_(p) 56(2)(1) to 56(2)(N−2) equal to the design choice value of the primary resistance R_(p), based on the first interconnection relationship 68, minus the fractional resistance R_(frac). The primary switch unit 42(2) is configured to include an adjacent combination of the at least one first fractional resistance R_(frac1) 58(2)(1) to 58(2)(N−1) to create a total resistance of primary resistance R_(p) 56(2)+the fractional resistance R_(frac) 58(2) in the first adjusting circuits 54(2)(i). The resistance value of the plurality of primary resistances R_(p) 56(2)(1) to 56(2)(N−2) will be adjusted where a resistance value substantially equal to that of the fractional resistance R_(frac) 58(2) is removed from the plurality of primary resistances R_(p) 56(2)(1) to 56(2)(N−2). The adjustment to the resistance value is necessary because the primary switch unit 42(2) will include the two immediately adjacent first fractional resistances R_(frac) 58(2) into the selected resistor node pair N_(r)(2)(1)-N_(r)(2)(N−2) 49. In this configuration, the value of the primary resistances R_(p) 56(2)(1) to 56(2)(N−2) have been reduced by shrinking their equivalent resistive units, first fractional resistances R_(frac) 58(2) are being reused were possible and additional first fractional resistance resistances have been eliminated saving physical space consumed by the circuit design.

For example, with continuing reference to FIG. 6, MSB code 48(2) has 3 bits so N=2³ or eight (8) and i is equal to the binary value of 100₂. The decimal equivalent of the binary value of 100₂ is four (4₁₀). In normalized resistive units R_(u), R_(p)=4R_(u), R_(frac)=0.5R_(u) and R_(sd)=36R_(u). The first interconnection relationship 68, R_(p)=R_(sd)∥(R_(p)+R_(frac)), will be met based on resolving the first interconnection relationship 68 with the values above 1/(1/(4R_(u)+0.5R_(u)))+1/36R_(u)))=4R_(u)=R_(p). The second interconnection relationship 72, (N-i-1)*R_(p)+R_(bulk2), is met if the total resistance between the voltage rail node V_(top) 44(2) and the selected resistor node pair N_(r)(2)(4) 49 equals (N-i-1)*R_(p)+R_(bulk2), where R_(bulk2) may equal zero. In this example, i=four (4), primary resistance R_(p) 56(2)(4)=3.5R_(u) and R_(bulk2)=zero (0). Based on the second interconnection relationship 72, (N-i-1)*R_(p)+R_(bulk2), the resistance between the voltage rail node V_(top) 44(2) and the selected resistor node pair must equal (8-4-1)*4R_(u)+0. This reduces to 3*4R_(u)+0, and further reduces to 12Ru or 3Rp. The resistances between the voltage rail node V_(top) 44(2) and the selected resistor node pair N_(r)(2)(4) 49(2) add up as follows: R_(p) 56(2)(5)+R_(p) 56(2)(6)+R_(p) 56(2)(7)+R_(frac) 58(2)(6)+R_(frac) 58(2)(7). With the resistance values inserted the formula reduces to 3.5R_(u)+3.5R_(u)+4R_(u)+0.5R_(u)+0.5 R_(u) which equals 12Ru, thus the second interconnection relationship 72 is met. The third interconnection relationship 74, i*R_(p)+R_(bulk1), where R_(bulk1) may equal to zero (0) reduces to 4*(R_(p))+0 or 4R_(p). Counting the resistance between the ground rail node V_(bot) 46(2) and the selected resistor node pair N_(r)(2)(4) 49(2) there are 4R_(u)+3*3.5R_(u)+3*0.5R_(u) or 16R_(u) or 4R_(p).

In a second example and with continued reference to FIG. 6, the first adjusting circuits 54(2)(0) and 54(2)(N−1) are configured differently from the previous first adjusting circuits 54(2)(1) and 54(2)(N−2), however, the all first adjusting circuits yield the same result. With all other parameters the same, except that MSB code 48(2) will now equal 000₂, the interconnection relationships 68, 72, 74 will continue to be met. The selected resistor node pair N_(r)(2)(0) 49(2) comprises the primary resistance R_(p) 56(2)(0)+the first fractional resistance R_(frac) 58(2)(1) which equals 4.5R_(u). The resistive value of 4.5R_(u) coupled in parallel to 36R_(u) equals 4R_(u) based on the parallel resistive equation as discussed above. The second interconnection relationship 72 is met and reduces to 6*3.5R_(u)+4R_(u)+6*0.5R_(u) further reducing, mathematically, to 28R_(u). 28Ru is the equivalent of seven (7) selectable resistor node pairs N_(r)(2)(i) 49(2) between the voltage rail node V_(top) 44(2) and the upper selected resistor node N_(r)(2)(0)H 50(2). The third interconnection relationship 74, i*R_(p)+R_(bulk1), where R_(bulk1) is equal to zero (0), reduces to 0*(R_(p))+0=0R_(u) between the ground rail node V_(bot) 46(2) and the lower selected resistor node N_(r)(2)(0)L 52(2). Since the selected resistor node pair N_(r)(2)(0) 49(2) is coupled to the ground rail node V_(bot) 46(2), the third interconnection relationship 74 is met. FIGS. 4 and 6 describe exemplary embodiments that comprise selecting a different resistor node pair 49 with a different primary resistance 56 with each unique value of i. It is also possible to eliminate switches in the primary voltage divider 30 by configuring the primary resistance R_(p) 56 and the fractional resistance R_(frac) 58 to remain constant. In this manner, instead of selecting different resistor node pairs 49, it may be desired to configure the adjusting circuits to reconfigure coupling of primary resistors. It may be desired to reconfigure coupling between a voltage rail node V_(top) 44(2) and the selected resistor node pair N_(r)(2)(i) 49(2), and a ground rail node V_(bot) 46(2) and the selected resistor node pair N_(r)(2)(i) 49(2), to maintain the ideal voltage of the selected resistor node pair N_(r)(2)(i) 49(2) across the secondary voltage divider 32(2), as discussed below.

In this regard, FIG. 7 illustrates an exemplary embodiment of a primary resistor string 40(3) comprising a primary resistance R_(p) 56(3) and at least one first fractional resistance R_(frac) 58(3). The primary switch unit 42(3) is configured to select a resistor node circuit 47(3) among a plurality of resistor node circuits 47, the resistor node circuit 47(3) comprises a selected resistor node pair N_(r)(3)(i) 49(3). The first fractional resistance R_(frac) 58(3) may also be referred to as a “shared fractional resistance”. The primary resistor string 40(3) in FIG. 7 is designed to operate based on the first, second, and third interconnection relationships 68, 72, 74 in FIG. 3, and included in the operation of the dual-string DACs 28 in FIGS. 4 and 6. However, the primary resistor string 40(3) is an alternative configuration comprising a plurality of adjusting primary resistances R_(p) _(—) _(adj)(3)(0) to R_(p) _(—) _(adj)(3)(N−2). The embodiment in FIG. 7 reconfigures the plurality of adjusting primary resistances R_(p) _(—) _(adj)(3)(0) to R_(p) _(—) _(adj)(3)(N−2) from between a voltage rail node V_(top) 44(3) and a higher selected resistor node N_(r)(3)(i)H 50(3) to between the ground rail node V_(bot) 46(3) and a lower selected resistor node N_(r)(3)(i)L 52(3) with each increment of the MSB code 48(3). In this manner, the voltages of the selected resistor node pair N_(r)(3)(i) 49(3) will increment by the corresponding number of coarse divided primary voltages. The primary resistance R_(p) 56(3) is serially coupled to the first fractional resistance R_(frac) 58(3). The serial coupling of the primary resistance R_(p) 56(3) and the first fractional resistance R_(frac) 58(3) is further coupled in parallel to the secondary resistor string 60(3). The parallel coupling will satisfy the first interconnection relationship 68, R_(p)=R_(sd)∥(R_(p)+R_(frac)). The second interconnection relationship 72 (N-i-1)*R_(p)+R_(bulk2) defines the resistance between the voltage rail node V_(top) 44(3) and the higher selected resistor node N_(r)(3)(i)H 50(3). The third interconnection relationship 74, i*R_(p)+R_(bulk1), defines the resistance between the ground rail node V_(bot) 46(3) and the lower selected resistor node N_(r)(3)(i)L 52(3).

With continuing reference to FIG. 7, the decoded output of the MSB code 48(3) will control a plurality of primary resistor string switches U(3)(0) to U(3)(3*N−1). The first N of the plurality of primary resistor string switches U(3)(0) to U(3)(N−1) are coupled between the plurality of adjusting primary resistances R_(p) _(—) _(adj)(3)(0) to R_(p) _(—) _(adj)(3)(N−2). A remaining plurality of primary resistor string switches U(3)(N) to U(3)(3*N−1) are coupled between the plurality of adjusting primary resistances R_(p) _(—) _(adj)(3)(0) to R_(p) _(—) _(adj)(3)(N−2) and the shared serial coupling of the primary resistance R_(p) 56(3) and the at least one first fractional resistance R_(frac) 58(3). Based on the decoded output of the MSB code 48(3), the remaining plurality of primary resistor string switches U(3)(N) to U(3)(3*N−1) will selectively couple the shared serial coupling of the primary resistance R_(p) 56(3) and the first fractional resistance R_(frac) 58(3) within the primary resistor string 40(3). This selective coupling based on the MSB code 48(3) will be configured according to the second interconnection relationship 72 and the third interconnection relationship 74 as discussed below.

In continuing reference to FIG. 7, FIG. 8A shows an example of operation where MSB code 48(3) has two (2) bits with a maximum value of N=2²=four (4) and for this example has a MSB code 48(3) value equal to 00₂ or in a decimal conversion i=zero (0₁₀). In normalized resistive units R_(u): R_(p)=4Ru, R_(frac)=0.5R_(u), R_(p) _(—) _(adj)(3)(0) to R_(p) _(—) _(adj)(3)(N−2)=R_(p), and R_(sd)=36R_(u). The first interconnection relationship 68, R_(p)=R_(sd)∥(R_(p)+R_(frac)), resolves to 1/(1/(4R_(u)+0.5R_(u))+(1/36R_(u)))=4R_(u)=R_(p). The second interconnection relationship 72, (N-i-1)*R_(p)+R_(bulk2), where R_(bulk2) is equal to zero (0), resolves to (4-0-1)*R_(p)+0 or 3R_(p). When i=zero (0), primary resistor string switches U(3)(0) U(3)(1) U(3)(2) U(3)(10) U(3)(11) are closed. These primary resistor string switch closures insert three (3) adjusting primary resistances R_(p) _(—) _(adj)(3)(0) to R_(p) _(—) _(adj)(3)(N−2) in series between the voltage rail node V_(top) 44(3) and the higher selected resistor node N_(r)(3)(0)H 50(3). This insertion of adjusting primary resistances R_(p) _(—) _(adj)(3)(0) to R_(p) _(—) _(adj)(3)(N−2) causes the second interconnection relationship 72 to be met. The third interconnection relationship 74, i*R_(p)+R_(bulk1), where R_(bulk1) is equal to zero (0), reduces to 0*R_(p)+0 or 0R_(p). This couples the lower selected resistor node N_(r)(3)(0)L 52(3) to the ground rail node V_(bot) 46(3) and the resistance is equal to 0R_(p), thus the third interconnection relationship 74 is met.

In continuing reference to FIG. 7, FIG. 8B shows an additional example of operation where MSB code 48(3) has 2 bits with a maximum value of N=2²=four (4) and for this example has a MSB code 48(3) value equal to 10₂ or decimally converted, i is equal to two (2₁₀). In normalized resistive units R_(u): R_(p) is equal to 4R_(u); R_(frac)=0.5R_(u); R_(p) _(—) _(adj)(3)(0) to R_(p) _(—) _(adj)(3)(N−2)=R_(p); and R_(sd)=36R_(u). The first interconnection relationship 68, R_(p)=R_(sd)∥(R_(p)+R_(frac)), resolves to 1/(1/(4R_(u)+0.5R_(u)))+1/36R_(u)))=4R_(u)=R_(p). The second interconnection relationship 72, (N-i-1)*R_(p)+R_(bulk2), where R_(bulk2) is equal to zero (0) resolves to (4-2-1)*R_(p)+0 or 1R_(p). When i=2, primary resistor string switches U(3)(0) U(3)(2) U(3)(3) U(3)(6) U(3)(7) are closed. This places one (1) R_(p) _(—) _(adj)(3)(2) between the voltage rail node V_(top) 44 ₍₃₎ and the higher selected resistor node N_(r)(3)(0)H 50(3), or 1R_(p), thus the second interconnection relationship 72 is met. The third interconnection relationship 74, i*R_(p)+R_(bulk1), where R_(bulk1) is equal to zero (0). Resolving the third interconnection relationship 74 equals 2*R_(p)+0 or 2R_(p). This inserts 2R_(p) between the ground rail node V_(bot) 46(3) and the lower selected resistor node N_(r)(3)(2)L 52(3), thus the third interconnection relationship 74 is met.

FIGS. 5-8 are exemplary embodiments using an adjusting circuit embedded within the primary resistor string 40 and more particularly, within the selected resistor node pairs N_(r)(i) 49. However, further embodiments are possible by introducing at least one additional adjusting circuit between the voltage rail node V_(top) 44 and the primary resistor string 40, or between the ground rail node V_(bot) 46 and the primary resistor string 40, or both. In this regard, FIG. 9 is a generalized approach of a dual-string DAC 28(4) with a configuration comprising at least one additional adjusting circuit. In this example, there is a primary voltage divider 30(4) coupled to a secondary voltage divider 32(4). The primary voltage divider 30(4) comprises a primary resistor string 40(4), a primary switch unit 42(4) and may include a second adjusting circuit 82(4) and/or a third adjusting circuit 83(4). The primary switch unit 42(4) is configured to select a resistor node circuit 47(4) among a plurality of resistor node circuits 47, the resistor node circuit 47(4) comprises a selected resistor node pair N_(r)(i) 49(4). The second adjusting circuit 82(4) is coupled between a voltage rail node V_(top) 44(4) and the primary resistor string 40(4). The third adjusting circuit 83(4) is coupled between a ground rail node V_(bot) 46(4) and the primary resistor string 40(4). The decoded MSB code 48(4) will determine the setting of the primary switch unit 42(4), and the necessary adjustments in the second adjusting circuit 82(4) and the third adjusting circuit 83(4). A decoded LSB code 66(4) will determine the setting for the secondary switch unit 64, or otherwise known as the “secondary voltage divider switch 64.”

In this regard, FIG. 10 is an illustration of an exemplary embodiment of a dual-string DAC 28(5) with three adjusting circuits. The dual-string DAC 28(5) of FIG. 10 comprises a primary voltage divider 30(5) and a secondary voltage divider 32(5). The primary voltage divider 30(5) comprises at least one first adjusting circuit 54(5)(1) to 54(5)(N−2), a second adjusting circuit 82(5), a third adjusting circuit 83(5) and a primary switch unit 42(5). The primary switch unit 42(5) is configured to select a resistor node circuit 47(5) among a plurality of resistor node circuits 47, the resistor node circuit 47(5) comprises a selected resistor node pair N_(r)(5)(i) 49(5). The plurality of the first adjusting circuits 54(5)(1) to 54(5)(N−2) comprises a primary resistance R_(p) 56(5)(1) to 56(5)(N−2) and at least one first fractional resistance R_(frac) 58(5)(1) to 58(5)(N−2). Each of the plurality of first adjusting circuits 54(5)(1) to 54(5)(N−2) are similarly configured to the first adjusting circuits 54(1) to 54(4) discussed above in FIGS. 5-9 comprising a total resistance (R_(p) 56(5)+R_(frac) 58(5)) to conform to the first interconnection relationship 68, R_(p)=R_(sd)∥(R_(p)+R_(frac)).

In this regard, the second adjusting circuit 82(5) is coupled between a voltage rail node V_(top) 44(5) and the lower selected resistor node N_(r)(5)(N−1)L 52(5). The second adjusting circuit 82(5) is comprised of a plurality of second fractional resistances R_(frac) 84(5)(1) to 84(5)(N−1) coupled in parallel to a plurality of second switches 86(5)(1) to 86(5)(N−1). The second adjusting circuit 82(5) is configured to add an additional one of the plurality of second fractional resistances R_(frac) 84(5)(1) to 84(5)(N−1) for each incremental increase in the MSB code 48(5) from zero (0) to N−1. The second adjusting circuit 82(5) does this by initially closing the plurality of second switches 86(5)(1) to 86(5)(N−1) and incrementally opening the plurality of second switches 86(5)(1) to 86(5)(N−1) as the MSB code 48(5) increments from 0 to N−1. In this manner, the second adjusting circuit 82(5) will compensate for any of the plurality of first fractional resistances R_(frac) 58(5)(1) to 58(5)(N−2) being removed from between the voltage rail node V_(top) 44(5) and the high selected resistor node N_(r)(5)(i)H 50(5) with each of the successively selected resistor node pairs N_(r)(5)(0) to N_(r)(5)(N−1) 49(5). By compensating for the first fractional resistance R_(frac) 58(5) changes, the total resistance of the primary resistor string 40(5) from the voltage rail node V_(top) 44(5) to the ground rail node V_(bot) 46(5) will remain substantially constant. The substantially constant total resistance prevents non-linearities when successively selecting the selected resistor node pairs N_(r)(5)(0) to N_(r)(5)(N−1) 49(5).

With continuing reference to FIG. 10, the third adjusting circuit 83(5) is coupled between the ground rail node V_(bot) 46(5) and the higher selected resistor node N_(r)(5)(0)H 50(5). As the MSB code 48(5) increases incrementally from zero (0) to N−1, the third adjusting circuit 83(5) compensates for the additional second fractional resistance R_(frac) 84(5) incrementally added in the second adjusting circuit 82(5). The third adjusting circuit 83(5) compensates by incrementally removing one of a plurality of third fractional resistances R_(frac) 88(5)(1) to 88(5)(N−1). The third adjusting circuit 83(5) performs the compensation by initially opening a plurality of third switches 90(5)(1) to 90(5)(N−1) when the MSB code 48(5) is zero (0). This will add the plurality of third fractional resistances R_(frac) 88(5)(1) to 88(5)(N−1) comprising the third adjusting circuit 83(5) between the selected resistor node pair N_(r)(5)(0) 49(5) and the ground rail node V_(bot) 46(5). The plurality of third switches 90(5)(1) to 90(5)(N−1) are incrementally closed as the MSB code 48(5) increments from zero (0) to N−1. The incremental closure of the third switches 90(5)(1) to 90(5)(N−1) will compensate for any of the additional first fractional resistances R_(frac) 58(5)(1) to 58(5)(N−2) that are added between the selected resistor node pair N_(r)(5)(i) 49(5) and the ground rail node V_(bot) 46(5). The primary voltage divider 30(5) is further configured to reverse this process as the MSB code 48(5) is incrementally decreased from N−1 to zero (0). The second adjusting circuit 82(5) removes one of the plurality of second fractional resistances R_(frac) 84(5) by incrementally closing a corresponding one of the plurality of second switches 86(5)(1) to 86(5)(N−1). Simultaneously, the third adjusting circuit 83(5) will incrementally add one of the plurality of third fractional resistances R_(frac) 88(5)(1) to 88(5)(N−1) by incrementally opening a corresponding one of the plurality of third switches 90(5)(1) to 90(5)(N−1).

In the example embodiment of FIG. 10, MSB code 48(5) has three (3) bits, N=2³=eight (8), R_(p)=4R_(u), R_(frac)=0.5R_(u), and R_(sd)=36R_(u). When MSB code 48(5)=i=zero (0), the primary switch unit 42(5) selects the selected resistor node pair N_(r)(5)(0) 49(5) comprising the third adjusting circuit 83(5). In this example, the selected resistor node pair N_(r)(5)(0) 49(5) will have a resistance of 4.5R_(u) which satisfies the first interconnection relationship 68, R_(p)=R_(sd)∥(R_(p)+R_(frac)), by reducing to (4R_(u)+0.5R_(u))∥36R_(u)=4R_(u). As discussed above, the second adjusting circuit 82(5) will incrementally add one of the plurality of second fractional resistances R_(frac) 84(5)(1) to 84(5)(N−1) to the primary resistor string 40(5) with each incremental increase in MSB code 48(5). Initially the second adjusting circuit 82(5) begins with the plurality of second switches 86(5)(1) to 86(5)(N−1) closed when i=zero (0). By incrementally opening one of the plurality of second switches 86(5)(1) to 86(5)(N−1), one of the plurality of second fractional resistances R_(frac) 84(5)(1) to 84(5)(N−1) are incrementally added to the primary resistor string 40(5). The second interconnection relationship 72, (N-i-1)*R_(p)+R_(bulk2), where R_(bulk2) equals zero (0), resolves to (8-0-1)*(4Ru)+0=28Ru between the voltage rail node V_(top) 44(5) and the higher selected resistor node N_(r)(5)(0)H 50(5). This will result in 28Ru between the voltage rail node V_(top) 44(5) and the higher selected resistor node N_(r)(5)(0)H 50(5), thus the second interconnection relationship 72 is met. The third interconnection relationship 74, i*R_(p)+R_(bulk1), where R_(bulk1) equals zero (0), resolves to 0*4R_(u)+0=0Ru between the ground rail node V_(bot) 46(5) and the lower selected resistor node N_(r)(5)(0)L 52(5). Since the lower selected resistor node N_(r)(5)(0)L 52(5) is coupled to ground, there is 0Ru between the ground rail node V_(bot) 46(5) and the lower selected resistor node N_(r)(5)(0)L 52(5), thus the third interconnection relationship 74 is met.

With continuing reference to FIG. 10, MSB code 48(5) increments from zero (0) to one (1), then i=one (1) and the selected resistor node pair N_(r)(5)(1) 49(5) comprises one of the plurality of first adjusting circuits 54(5)(1). In this example, the selected resistor node pair N_(r)(5)(1) 49(5) will have a resistance of 4.5R_(u) which satisfies the first interconnection relationship 68, R_(p)=R_(sd)∥(R_(p)+R_(frac)), by reducing to (4R_(u)+0.5R_(u))∥36R_(u)=4R_(u). The second adjusting circuit 82(5) is configured to incrementally add one of the plurality of second fractional resistances R_(frac) 84(5)(1) to the primary resistor string 40(5). The purpose for this is to compensate for the first fractional resistance 58 that is being removed from the primary resistor string 40(5) with each successive increment of the MSB code 48(5). One of the plurality of second fractional resistances R_(frac) 84(5)(1) is incrementally added to the primary resistor string 40(5) between the voltage rail node V_(top) 44(5) and the higher selected resistor node N_(r)(5)(1)H 50(5). The incremental addition is a result of an increase in MSB code 48(5) and the opening of a corresponding one of the plurality of second switches 86(5)(1). The second interconnection relationship 72, (N-i-1)*R_(p)+R_(bulk2), resolves to (8-1-1)*4R_(u)+0=24R_(u). The resistance between the voltage rail node V_(top) 44(5) and the higher selected resistor node N_(r)(5)(1)H 50(5) is 24R_(u). With the incremental increase in MSB code 48(5), the third adjusting circuit 83(5) is also configured to incrementally remove one of the plurality of third fractional resistances R_(frac) 88(5)(1) from the primary resistor string 40(5) by closing one of the plurality of third switches 90(5)(1). This causes the third interconnection relationship 74, i*Rp+Rbulk1, to resolve to 1*4R_(u)+0=4R_(u) between the ground rail node V_(bot) 46(5) and the lower selected resistor node N_(r)(5)(i)L 52(5). The total resistance between the ground rail node V_(bot) 46(5) and the lower selected resistor node N_(r)(5)(1)L 52(5) is 4R_(u). This example embodiment introduces the technique of using a combination of first, second, and third adjusting circuits 54, 82, 83 to adjust the resistances such that the primary voltage divider 30(5) maintains a linear transfer function through each successive resistor node pair N_(r)(5)(i) 49(5). It is also possible to combine the exemplary embodiments creating a hybrid embodiment. The hybrid may use a combination of first adjusting circuits 54, second adjusting circuits 82 and third adjusting circuits 83 where fractional resistances R_(frac) 58 are shared by adjacent selected resistor node pairs 49, thus further reducing the number of switches and resistances.

In this regard, FIG. 11 is a hybrid of the exemplary embodiments in FIGS. 6 and 10. The exemplary embodiment of FIG. 11 uses the first, second, and third adjusting circuits 54, 82, 83 to maintain the first, second, and third interconnection relationships 68, 72, 74 as discussed above in FIG. 10. The primary switch unit 42(6) is configured to select a resistor node circuit 47(6) among a plurality of resistor node circuits 47, the resistor node circuit 47(6) comprises a selected resistor node pair N_(r)(6)(i) 49(6). In addition, the primary switch unit 42(6) is configured such that adjacent and overlapping resistor node pairs 49 defined by resistor node pairs N_(r)(6)(i) 49(6) are able to combine and share resistances as described in FIG. 6. As a result of the overlapping technique, this embodiment is able to also reduce the number of the second fractional resistances R_(frac) 84(6) and second switches 86(6) in a second adjusting circuit 82(6). The number of third fractional resistances R_(frac) 88(6) and third switches 90(6) in a third adjusting circuit 83(6) may also be reduced.

The exemplary embodiment of FIG. 11 comprises a primary resistor string 40(6) and the primary switch unit 42(6). The primary switch unit 42(6) is configured such that the primary switch unit 42(6) selects a selected resistor node pair N_(r)(6)(0) to N_(r)(6)(N−1) 49(6) based on a decoded MSB code 48(6). The resistance of the selected resistor node pair N_(r)(6)(0) to N_(r)(6)(N−1) 49(6) equals a design choice value primary resistance R_(p)(6)+fractional resistance R_(frac)(6). This exemplary embodiment may use resistances from adjacent adjusting circuits to create a total selected resistor node pair resistance of R_(p)(6)+R_(frac)(6). Where the R_(p)(6) and R_(frac)(6) resistance values are a design choice such that R_(p)(6)=R_(sd)(6)∥(R_(p)(6)+R_(frac)(6)). The primary switch unit 42(6) is also configured to combine a plurality of adjacent resistances such that the combination of adjacent resistances conform to the first, second, and third interconnection relationships 68, 72, 74. The primary resistor string 40(6) comprises at least one first adjusting circuit 54(6)(1) to 54(6)(N−2), a second adjusting circuit 82(6), and a third adjusting circuit 83(6). The second adjusting circuit 82(6) is coupled between a voltage rail node V_(top) 44(6) and the lower selected resistor node N_(r)(6)(N−1)L 52(6). The second adjusting circuit 82(6) is comprised of a plurality of second fractional resistances R_(frac) 84(6)(1) to 84(6)(X), a plurality of second switches 86(6)(1) to 86(6)(X) and a second adjusting resistance 92(6). Where X is a design choice dependent on a combination of resistor values in both the primary voltage divider 30(6), the secondary voltage divider 32(6) and may depend on the number of selectable resistor node pairs N_(r)(6)(0)-N_(r)(6)(N−1) 49(6). The third adjusting circuit 83(6) is coupled to a ground rail node V_(bot) 46(6) and the higher selected resistor node N_(r)(6)(0)H 50(6). The third adjusting circuit 83(6) is comprised of a plurality of third fractional resistances R_(frac) 88(6)(1) to 88(6)(Y), a plurality of third switches 90(6)(1) to 90(6)(Y) and a third adjusting resistance 94(6). Where Y is a design choice dependent on a combination of resistor values in both the primary voltage divider 30(6), the secondary voltage divider 32(6) and may depend on the number of selectable resistor node pairs N_(r)(6)(0)-N_(r)(6)(N−1) 49(6). The primary switch unit 42(6) is configured to include a combination of adjacent first fractional resistances R_(frac) 58 for a total resistance of R_(p)+R_(frac), such that the first interconnection relationship 68, R_(p)=R_(sd)∥(R_(p)+R_(frac)) in each of the first adjusting circuits 54(6)(1) to 54(6)(N−2) is satisfied. The second adjusting circuit 82(6) is configured to incrementally add or remove second fractional resistances R_(frac) 84(6) that will conform to the second interconnection relationship 72, (N-i-1)*R_(p)+R_(bulk2), where R_(bulk2) may be zero (0). The third adjusting circuit 83(6) is configured to incrementally remove or add third fractional resistances R_(frac) 88(6) that will conform to the third interconnection relationship 74, i*R_(p)+R_(bulk1), where R_(bulk1) may be zero (0). In this configuration, because of the reuse of the resistive units in contiguous selected resistor node pairs N_(r)(6) 49(6) the size of the resistances throughout the primary resistor string 40(6) may be reduced to be a fraction of the primary resistance R_(p) 56 in previous embodiments.

In the exemplary embodiment of FIG. 11, the MSB code 48(6) has three (3) bits, N=2³=eight (8), R_(p)=4R_(u), R_(frac)=0.5R_(u), and R_(sd)=36R_(u), R_(bulk1)=R_(bulk2)=0, when MSB code 48(6)=i=zero (0₁₀) the selected resistor node pair will be N_(r)(6)(0) 49(6), which will comprise the third adjusting circuit 83(6). In this example, the third adjusting circuit 83(6) comprising the plurality of third switches 90(6)(0) to 90(6)(Y) are all open, adding the plurality of third fractional resistances R_(frac) 88(6)(0) to 88(6)(Y) into the selected resistor node pair N_(r)(6)(0) 49(6). The resistance in the selected resistor node pair N_(r)(6)(0) 49(6) will have a total resistance of 4.5R_(u) which satisfies the first interconnection relationship 68, R_(p)=R_(sd)∥(R_(p)+R_(frac)), or (4R_(u)+0.5R_(u))∥36R_(u)=4R_(u). The second adjusting circuit 82(6) is configured to close the one of the plurality of second switches 86(6)(0), thus removing the plurality of second fractional resistances R_(frac) 84(6)(0) to 84(6)(X) from the primary resistor string 40(6). A number of resistive units R_(u) coupled between the voltage rail node V_(top) 44(6) and the higher selected resistor node N_(r)(6)(0)H 50(6) is 28R_(u). The second interconnection relationship 72, (N−i−1)*R_(p)+R_(bulk2), resolves to (8-0-1)*4R_(u)+0=28R_(u). A resistance of 28Ru is the correct resistance between the voltage rail node V_(top) 44(6) and the higher selected resistor node N_(r)(6)(0)H 50(6). The third adjusting circuit 83(6) is configured to open the plurality of third switches 90(6)(0) to 90(6)(Y), thus adding the plurality of third fractional resistances R_(frac) 88(6)(0) to 88(6)(Y) to the primary resistor string 40(6). Since the lower selected resistor node Nr(6)(0)L 52(6) is coupled to the ground rail node V_(bot) 46(6), the number of resistive units R_(u) coupled between the ground rail node V_(bot) 46(6) and the lower selected resistor node N_(r)(6)(0)L 52(6) is 0R_(u). The third interconnection relationship 74, i*R_(p)+R_(bulk1), resolves to (8-0-1)*4R_(u)+0=0R_(u) which, in this first example, is the resistance between the ground rail node V_(bot) 46(6) and the lower selected resistor node N_(r)(6)(0)L 52(6).

In continuing reference to FIG. 11, a second example for setting the MSB code 48(6) to 011₂ which converts decimally to i equal to three (3₁₀), is provided. All other settings remaining constant will result in the primary switch unit 42(6) selecting the selected resistor node pair N_(r)(6)(3) 49(6). The resistance of the selected resistor node pair N_(r)(6)(3) 49(6) is 2R_(u)+2.5R_(u) or 4.5R_(u). The first interconnection relationship 68 remains the same and 36R_(u)∥(4R_(u)+0.5R_(u))=4Ru. The second interconnection relationship 72 resolves to (8-3-1)*4R_(u)+0=16R_(u). The second adjusting circuit 82(6) is configured to open the plurality of second switches 86(6). Opening the plurality of second switches 86(6) adds 1.5Ru to the primary resistor string 40(6) for a total resistance between the voltage rail node V_(top) 44(6) and the higher selected resistor node N_(r)(6)(3)H 50(6) of 16R_(u). The third interconnection relationship 74 resolves to 3*4R_(u)+0=12R_(u). The third adjusting circuit 83(6) is configured to close one of the plurality of third switches 90(6)(3). Closing one of the plurality of third switches 90(6)(3) removes 1.5Ru from the primary resistor string 40(6) for a total resistance between the ground rail node V_(bot) 46(6) and the lower selected resistor node N_(r)(6)(3)L 52(6) of 12R_(u). This hybrid embodiment leverages adjacent resistances and in this manner also reduces the number of switches and resistances in the second adjusting circuit 82 and third adjusting circuit 83.

In this regard, FIG. 12 is an exemplary embodiment which allows at least one first adjusting circuit 54(7)(0) to 54(7)(N−1) to further reduce the number of switches and resistances by allowing several of the plurality of first adjusting circuits 54(7) to be completely identical for a plurality of unique MSB codes 48(7). That is, this embodiment has selectable resistor node pairs N_(r)(6)(i) 49(6) that are identical for a plurality of unique MSB codes 48(7) (not shown). This has the advantage of reducing by nearly half the number of resistive units and switches required to construct the primary resistor string 40(7). In this exemplary embodiment, a primary voltage divider 30(7) comprises a primary resistor string 40(7), a primary switch unit 42(7), at least one first adjusting circuit 54(7)(0) to 54(7)(N−1), a second adjusting circuit 82(7), and a third adjusting circuit 83(7). The second adjusting circuit 82(7) is coupled between a voltage rail node V_(top) 44(7) and the primary resistor string 40(7). The third adjusting circuit 83(7) is coupled between a ground rail node V_(bot) 46(7) and the primary resistor string 40(7). The primary switch unit 42(7) coupled to the primary resistor string 40(7) is configured to select a resistor node circuit 47(7) among a plurality of resistor node circuits 47, the resistor node circuit 47(7) comprises a selected resistor node pair N_(r)(7)(i) 49(7).

In FIG. 12, each one of the plurality of first adjusting circuits 54(7)(0) to 54(7)(N−1) are comprised of a resistance value equaling R_(p)(7)+R_(frac)(7). The resistance value of R_(p)(7)+R_(frac)(7) allows each one of the plurality of first adjusting circuits 54(7)(0) to 54(7)(N−1) to meet the first interconnection relationship 68, R_(p)=R_(sd)∥(R_(p)+R_(frac)). The second adjusting circuit 82(7) comprises a plurality of second switches 86(7)(0) to 86(7)(X) configured to add or remove a corresponding one of the plurality of second fractional resistances R_(frac) 84(7)(0) to 84(7)(X) to the primary resistor string 40(7) based on the MSB code 48(7). Adding or removing one of the plurality of second fractional resistances R_(frac) 84(7)(0) to 84(7)(X) enables the second adjusting circuit 82(7) to meet the second interconnection relationship 72, (N-i-1)*R_(p)+R_(bulk2) resistive units between the voltage rail node V_(top) 44(7) and a higher selected resistor node N_(r)(7)(i)H 50(7).

With continuing reference to FIG. 12, the third adjusting circuit 83(7) comprises a plurality of third switches 90(7)(0) to 90(7)(Y) configured to remove or add a corresponding one of the plurality of third fractional resistances R_(frac) 88(7)(0) to 88(7)(Y) between the ground rail node V_(bot) 46(7) and a lower selected resistor node N_(r)(7)(i)L 52(7) based on the MSB code 48(7). Adding or removing one of the plurality of third fractional resistances R_(frac) 88(7)(0) to 88(7)(Y) enables the third adjusting circuit 83(7) to meet the third interconnection relationship 74, i*R_(p)+R_(bulk1) resistive units between the ground rail node V_(bot) 46(7) and the lower selected resistor node N_(r)(7)(i)L 52(7). Adding or removing one of the plurality of second fractional resistances R_(frac) 84(7)(0) to 84(7)(X) and one of the plurality of third fractional resistances R_(frac) 88(7)(0) to 88(7)(Y) compensates for the changing of the selected resistor node pair N_(r)(7)(0) to N_(r)(7)(N−1) 49(7). As the equivalent of the first fractional resistances 58 R_(frac) are removed from between the voltage rail node V_(top) 44(7) and the higher selected resistor node N_(r)(7)(i)H 50(7), it is necessary for the second adjusting circuit 82(7) to add another fractional resistance 58 R_(frac) into the primary resistor string 40(7). It is also necessary to remove a fractional resistance 58 R_(frac) from the primary resistor string 40(7) between the ground rail node V_(bot) 46(7) and the lower selected resistor node N_(r)(7)(i)L 52(7) as the MSB code 48(7) is incremented. The adding or removing of fractional resistances 58 R_(frac) is because of the additional first fractional resistance 58 from the previously selected resistor node pair 49 that has just been added between the ground rail node V_(bot) 46(7) and the lower selected resistor node N_(r)(7)(i)L 52(7).

With regard to FIG. 12, an exemplary embodiment is shown with the following settings: the MSB code 48(7) has four (4) bits with N=2⁴=sixteen (16), R_(p)=3R_(u), R_(frac)=1R_(u), and R_(sd)=12R_(u), R_(bulk1)=R_(bulk2)=0. When MSB code 48(7) equals 0000₂, converting decimally to i equal to zero (0₁₀), this will cause the primary switch unit 42(7) to select a selected resistor node pair N_(r)(7)(0) 49(7). In this example, the selected resistor node pair N_(r)(7)(0) 49(7) will comprise the third adjusting circuit 83(7) comprising the plurality of third switches 90(7)(0) to 90(7)(Y). Y is a function of a combination of the resistor value design choice and number of selectable resistor node pairs N_(r)(7)(0) to N_(r)(7)(N−1) 49(7). One of the plurality of third switches 90(7)(1) is closed thereby coupling the lower selected resistor node N_(r)(7)(0)L 52(7) to the ground rail node V_(bot) 46(7), and one of the plurality of third switches 90(7)(Y) is opened. Opening one of the plurality of third switches 90(7)(Y) adds one of the plurality of the third fractional resistances R_(frac) 88(7)(Y) to the selected resistor node pair N_(r)(7)(0) 49(7). The resistance between the selected resistor node pair N_(r)(7)(0) 49(7) will have a total resistance of 4R_(u) which satisfies the first interconnection relationship 68, R_(p)=R_(sd)∥(R_(p)+R_(frac)) or 3R_(u)=12R_(u)∥(3R_(u)+1R_(u)). The second adjusting circuit 82(7) is configured to close one of the plurality of second switches 86(7)(X) and open two (2) of the plurality of second switches 86(7)(0) to 86(7)(1). In this manner, the second adjusting circuit 82(7) removes one of the plurality of second fractional resistances R_(frac) 84(7)(X) from the primary resistor string 40(7) and adds two (2) of the plurality of second fractional resistances R_(frac) 84(7)(0) to 84(7)(1) to the primary resistor string 40(7). The second interconnection relationship 72, (N-i-1)*R_(p)+R_(bulk2), resolves to (16-0-1)*3R_(u)+0=45R_(u), which is the resistance between the voltage rail node V_(top) 44(7) and the higher selected resistor node N_(r)(7)(0)H 50(7). The number of resistive units R_(u) coupled between the voltage rail node V_(top) 44(7) and the higher selected resistor node N_(r)(7)(0)H 50(7) is 45Ru and meets the second interconnection relationship 72.

As discussed above, in continuing reference to FIG. 12, the selected resistor node pair N_(r)(7)(0) 49(7) comprises the third adjusting circuit 83(7), however it is still necessary to meet the third interconnection relationship 74. In this manner, the third adjusting circuit 83(7) is configured to close one of the plurality of third switches 90(7)(1) and open two (2) of the plurality of third switches 90(7)(0) and 90(7)(Y). This configuration removes two (2) of the plurality of third fractional resistances R_(frac) 88(7)(0) to 88(7)(1) and adds one of the plurality of third fractional resistances R_(frac) 88(7)(Y) to the primary resistor string 40(7). In this manner, the third interconnection relationship 74, i*R_(p)+R_(bulk1) resolves to 0*3R_(u)+0=0R_(u), which is the resistance between the ground rail node V_(bot) 46(7) and the lower selected resistor node N_(r)(7)(0)L 52(7). Since the lower selected resistor node N_(r)(7)(0)L 52(7) is coupled to the ground rail node V_(bot) 46(7), the number of resistive units R_(u) is zero, thus meeting the third interconnection relationship 74.

With continuing reference to FIG. 12, a second example setting the MSB code 48(7)=i=one (1) and keeping all other settings constant will result in the primary switch unit 42(7) selecting the selected resistor node pair N_(r)(7)(1) 49(7). The resistance between selected resistor node pair N_(r)(7)(1) 49(7) is 4R_(u). The first interconnection relationship 68 remains the same as in the example above where the MSB code 48(7) equals zero (0) and is satisfied by the parallel equation R_(p)=R_(sd)∥(R_(p)+R_(frac)) or (3R_(u)+1R_(u))∥12R_(u)=3R_(u). The second interconnection relationship 72 resolves to (16-1-1)*3R_(u)+0=42R_(u). The second adjusting circuit 82(7) is configured to open the plurality of second switches 86(7)(0) to 86(7)(X), thus adding 3R_(u) to the primary resistor string 40(7) between the voltage rail node V_(top) 44(7) and the higher selected resistor node N_(r)(7)(1)H 50(7) of 42R_(u). The third interconnection relationship 74 resolves to 1*3R_(u)+0=3R_(u). The third adjusting circuit 83(7) is configured to close one of the plurality of third switches 90(7)(Y), thus removing 3R_(u) from the primary resistor string 40(7) for a total resistance between the ground rail node V_(bot) 46(7) and the lower selected resistor node N_(r)(7)(1)L 52(7) of 3R_(u). This hybrid embodiment may leverage the same selectable resistor node pairs 49 for unique instances of the MSB code 48(7) based on an alternative configuration of the second adjusting circuit 82(7) and the third adjusting circuit 83(7).

In this regard, and in continuing reference to FIG. 12, when MSB code 48(7) is equal to two (2), the selected resistor node pair N_(r)(7)(2) 49(7) will remain the same as the selected resistor node pair N_(r)(7)(1) 49(7) discussed in detail above even with a different MSB code 48(7). The difference is in the second adjusting circuit 82(7) configuration which closes one of the plurality of second switches 86(7)(0) that will remove the plurality of second fractional resistances R_(frac) 84(7)(0) to 84(7)(X) from the primary resistor string 40(7). The removal of the plurality of second fractional resistances R_(frac) 84(7)(0) to 84(7)(X) has the effect of removing 3R_(u) or 1R_(p) from between the voltage rail node V_(top) 44(7) and the higher selected resistor node N_(r)(7)(2)H 50(7). The third adjusting circuit 83(7) is configured to open the plurality of third switches 90(7)(0) to 90(7)(Y), adding the plurality of third fractional resistances R_(frac) 88(7)(0) to 88(7)(Y) to the primary resistor string 40(7). The adding of the plurality of third fractional resistances R_(frac) 88(7)(0) to 88(7)(Y) has the effect of adding 3R_(u) or 1R_(p) between the ground rail node V_(bot) 46(7) and the lower selected resistor node N_(r)(7)(2)L 52(7). Fractional resistances R_(frac) are being reconfigured from between the voltage rail node V_(top) 44(7) and the higher selected resistor node N_(r)(7)(1)H 50(7) to between the ground rail node V_(bot) 46(7) and the lower selected resistor node N_(r)(7)(2)L 52(7). The adding and removal of fractional resistances R_(frac) has the effect of increasing the selected resistor node pair N_(r)(7)(2) 49(7) by one (1) coarse voltage division and yet maintaining the same physically selected resistor node pair N_(r)(7)(1) 49(7) as when the MSB code 48(7) equals one (1). The number of required fractional resistances R_(frac) used in the second adjusting circuit 82(7) and the third adjusting circuit 83(7) will depend on design choices. The design choices for the resistance values may be made by the designer based for example on area and function, as non-limiting examples. While the exemplary embodiment of FIG. 12 shows a plurality of second fractional resistances R_(frac) 84(7) and a plurality of third fractional resistances R_(frac) 88(7) comprising three (3) resistances, it is possible to even further reduce the number of fractional resistances in the adjusting circuits 54, 82, 83.

In this regard, FIG. 13 uses a design choice of the MSB code 48(8) having five (5) bits with N=2⁵=thirty-two (32), R_(p)=1R_(u), R_(frac)=1R_(u), and R_(sd)=2R_(u), R_(bulk1)=R_(bulk2)=0. This exemplary embodiment is comprised of a primary resistor string 40(8), a primary switch unit 42(8), at least one first adjusting circuit 54(8)(0) to 54(8)(N−1), a second adjusting circuit 82(8), and a third adjusting circuit 83(8). The primary switch unit 42(8) is configured to select a resistor node circuit 47(8) among a plurality of resistor node circuits 47, the resistor node circuit 47(8) comprises a selected resistor node pair N_(r)(8)(i) 49(8). The selected resistor node pair Nr(8)(i) 49(8) is selected based on the corresponding MSB code 48(8) associated with the selected resistor node pair 49(8) in FIG. 13. The switches U1 and U2 are closed for the indicated codes. The first, second, and third interconnection relationship 68, 72, 74 equations remain the same as discussed above. In this example, the secondary serial resistance R_(sd)=2R_(u), which allows the primary resistance R_(p)=fractional resistance R_(frac)=resistive unit R_(u), is based on the first interconnection relationship 68, R_(p)=R_(sd)∥(R_(p)+R_(frac)). As a result of R_(p)=R_(frac)=R_(u), the second adjusting circuit 82(8) may comprise a single second fractional resistance R_(frac) 84(8), and the third adjusting circuit 83(8) may comprise a single third fractional resistance R_(frac) 88(8). The first interconnection relationship 68, R_(p)=R_(sd)∥(R_(p)+R_(frac)) resolves to (1R_(u)+1R_(u))∥2R_(u)=1R_(u). The first adjusting circuit 54(8)(0) in the selected resistor node pair N_(r)(8)(0) 49(8) comprises 2R_(u) or (R_(p)+R_(frac)).

With continuing reference to FIG. 13, when MSB code 48(8) is equal to zero (0), the second adjusting circuit 82(8) opens a single second switch 86(8) adding the single second fractional resistance R_(frac) 84(8) between the voltage rail node V_(top) 44(8) and the higher selected resistor node N_(r)(8)(0)H 50(8). The adding of a second fractional resistance R_(frac) 84(8) creates a total resistance between the voltage rail node V_(top) 44(8) and the higher selected resistor node N_(r)(8)(0)H 50(8) of 31R_(u). The second interconnection relationship 72, (N-i-1)*R_(p)+R_(bulk2) resolves to (32-0-1)*1R_(u)+0=31R_(u). The third interconnection relationship 74, i*R=R_(p)+R_(bulk1) resolves to 0*1R_(u)+0=0R_(u). Since the lower selected resistor node N_(r)(8)(0)L 52(8) is coupled to the ground rail node V_(bot) 46(8), the resistance between them is equal to 0R_(u).

Incrementing the MSB code 48(8) of FIG. 13 by one (1) so MSB code 48(8)=i=1, the first adjusting circuit 54(8)(1) for the selected resistor node pair N_(r)(8)(1) 49(8) will remain the same as previously selected resistor node pair N_(r)(8)(0) 49(8). However, as was discussed in greater detail in FIG. 12, the single second fractional resistance R_(frac) 84(8) in the second adjusting circuit 82(8) is removed from between the voltage rail node V_(top) 44(8) and the higher selected resistor node N_(r)(8)(0)H 50(8) by closing the single second switch 86(8). The third adjusting circuit 83(8) adds a single third fractional resistance R_(frac) 88(8)(0) in the third adjusting circuit 83(8) between the ground rail node V_(bot) 46(8) and the lower selected resistor node N_(r)(8)(1)L 52(8) by opening a single third switch 90(8). This has the circuit equivalent effect of reconfiguring the second fractional resistance 84(8) from the second adjusting circuit 82(8) to the third adjusting circuit 83(8). In this manner, the first, second, and third interconnection relationship 68, 72, 74 equations are met, and a linear output voltage is maintained on the selected resistor node pairs N_(r)(8)(0) to N_(r)(8)(N−1) 49(8). This exemplary embodiment and all of the previously discussed primary voltage dividers 30 are voltage sourced driven. The primary voltage divider 30(1) to 30(8) then divides the voltage between the voltage rail node V_(top) 44(1) to 44(8) and the ground rail node V_(bot) 46(1) to 46(8).

FIG. 14 illustrates an exemplary embodiment of a dual-string DAC 28, which operates in a similar fashion to the embodiments described above in FIGS. 12 and 13. The exemplary embodiment in FIG. 14, will function in conformance with the first, second, and third interconnection relationships 68, 72, 74 as described above. FIG. 14 uses a non-limiting design choice of the MSB code 48(9) having five (5) bits with N=2⁵=thirty-two (32), R_(p)=1R_(u), R_(frac)=1R_(u), and R_(sd1)−R_(sd2)=2R_(u), R_(bulk1)=R_(bulk2)=0. This exemplary embodiment is comprised of a primary resistor string 40(9), a primary switch unit 42(9), at least one first adjusting circuit 54(9)(0) to 54(9)(N−1), a second adjusting circuit 82(9), and a third adjusting circuit 83(9). The primary switch unit 42(9) is configured to select a resistor node circuit 47(9) among a plurality of resistor node circuits 47, the resistor node circuit 47(9) comprises a selected resistor node pair N_(r)(9)(i) 49(9). The selected resistor node pair Nr(9)(i) 49(9) is selected based on the corresponding MSB code 48(9) associated with the selected resistor node pair 49(9). The switches U1 to U12 are closed for the indicated MSB codes 48(9). The first, second, and third interconnection relationship 68, 72, 74 equations remain the same as discussed the embodiments above. In this example, the secondary serial resistance R_(sd2)=R_(sd1)=2R_(u), which allows the primary resistance R_(p)=fractional resistance R_(frac)=resistive unit R_(u). The value of the primary resistance R_(p)=fractional resistance R_(frac)=resistive unit R_(u) is based on the first interconnection relationship 68, R_(p)=R_(sd)∥(R_(p)+R_(frac)). The first interconnection relationship 68, R_(p)=R_(sd)∥(R_(p)+R_(frac)) resolves to (1R_(u)+1R_(u))∥2R_(u)=1R_(u). The first adjusting circuit 54(9)(0) in the selected resistor node pair N_(r)(9)(0) 49(9) comprises 2R_(u) or (R_(p)+R_(frac)).

With continuing reference to FIG. 14, when MSB code 48(9) is equal to zero (0), the second adjusting circuit 82(9) closes the corresponding switch U1 and opens the remaining switches in the second adjusting circuit 82(9). In this manner, the second adjusting circuit 82(9) adds 8Ru between the voltage rail node V_(top) 44(9) and the higher selected resistor node N_(r)(9)(0)H 50(9). It is important to note that the resistor R8 is 2Ru and has a parallel resistance R_(sd2) of 2Ru. The total parallel resistance at resistor R8 is 1Ru. The adding of the second fractional resistances R_(frac) 84(9) creates a total resistance between the voltage rail node V_(top) 44(9) and the higher selected resistor node N_(r)(9)(0)H 50(9) of 31R_(u). The second interconnection relationship 72, (N-i-1)*R_(p)+R_(bulk2) resolves to (32-0-1)*1R_(u)+0=31R_(u). The third interconnection relationship 74, i*R=R_(p)+R_(bulk1) resolves to 0*1R_(u)+0=0R_(u). Since the lower selected resistor node N_(r)(9)(0)L 52(9) is coupled to the ground rail node V_(bot) 46(9), the resistance between them is equal to 0R_(u).

With continuing reference to FIG. 14, this embodiment provides for a secondary voltage divider represented by a plurality of secondary voltage dividers. As a non-limiting example, FIG. 14 will show an exemplary two (2) secondary voltage dividers represented by the secondary resistor string R_(sd1) and R_(sd2) 60(9). It may be possible for the embodiments in FIGS. 4-14 to comprise a plurality of secondary voltage dividers as may be necessary based on design choices, material limitations and construction techniques, etc. Which secondary resistor string, R_(sd1) or R_(sd2) 60(9), is coupled to the DAC output voltage V_(out) 38(9) will be based on the MSB code 48(9) that is input into the primary switch unit 42(9). As in the example above, if the MSB code 48(9) is zero (0) the primary switch unit 42(9) will open switch U11 and close switch U12. Opening the switch U11 will not remove the secondary resistor string R_(sd2) 60(9) from being coupled to the resistor node pair at R8. However, by closing switch U12, the DAC output voltage V_(out) 38(9) will be provided by the secondary resistor string R_(sd1) 60(9) through the coupling by the switch U12. Further, the secondary resistor string R_(sd1) will also be coupled to the corresponding selected resistor node pair Nr(9)(i) 49(9) based on the MSB code 48(9). If the MSB code 48(9) were incremented to twelve (12), the secondary resistor string R_(sd1) 60(9) would be decoupled from both the DAC output voltage V_(out) 38(9) and the primary resistor string 40(9). In addition, based on the MSB code 48(9)=twelve (12₁₀) the secondary resistor string R_(sd2) 60(9) is coupled to DAC output voltage Vout 38(9).

Incrementing the MSB code 48(9) of FIG. 14 by one (1) will set i=1. The first adjusting circuit 54(9)(1) for the selected resistor node pair N_(r)(9)(1) 49(9) will remain the same as previously selected resistor node pair N_(r)(9)(0) 49(9). The third adjusting circuit 83(9) adds a single third fractional resistance R_(frac) 88(9)(0) in the third adjusting circuit 83(9) between the ground rail node V_(bot) 46(9) and the lower selected resistor node N_(r)(9)(1)L 52(9) by opening a third switch U2 90(9)(1). In this manner, the first, second, and third interconnection relationship 68, 72, 74 equations are met, and a linear output voltage is maintained on the selected resistor node pairs N_(r)(9)(0) to N_(r)(9)(N−1) 49(9). This exemplary embodiment and all of the previously discussed primary voltage dividers 30 are voltage sourced driven. The primary voltage divider 30(1) to 30(9) then divides the voltage between the voltage rail node V_(top) 44(1) to 44(9) and the ground rail node V_(bot) 46(1) to 46(9). However, instead of driving the primary voltage divider 30 by a voltage source, it is also possible to drive the primary voltage divider 30 with a current source. The voltages across the selected resistor node pair 49 will become a function of the current multiplied by the resistance.

In this regard, FIG. 15 illustrates an exemplary embodiment of a dual-string DAC 28 with a current sourced primary voltage divider 30(10). The primary voltage divider 30(10) comprises at least one current source 96, a primary resistor string 40(10), a third adjusting circuit 83(10), and a primary switch unit 42(10). The primary switch unit 42(10) is configured to select a resistor node circuit 47(10) among a plurality of resistor node circuits 47, the resistor node circuit 47(10) comprises a selected resistor node pair N_(r)(i) 49(10). The at least one current source 96 is coupled to the primary resistor string 40(10) at a higher selected resistor node N_(r)(10)(N−1)H 50(10). The at least one current source 96 coupled to the primary resistor string 40(10) may optionally be coupled to a trim resistor 97. The resistance of the trim resistor 97, if present, is adjusted to maintain a constant current at Vtop 44(10), The third adjusting circuit 83(10) is coupled between the ground rail node V_(bot) 46(10) and the primary resistor string 40(10) at the lower selected resistor node N_(r)(10)(0)L 52(10). While this is a current sourced primary voltage divider 30(10), the operation is similar to the voltage sourced primary voltage dividers 30 of FIGS. 9-13. In an alternative embodiment, the at least one current source 96 and the third adjusting circuit 83(10) would switch positions in the primary voltage divider 30(10). In the alternative embodiment, the at least one current source 96 would be coupled between the ground rail node V_(bot) 46(10) and the lower selected resistor node N_(r)(10)(0)L 52(10). The third adjusting circuit 83(10) would then be coupled between the voltage rail node V_(top) 44(10) and the higher selected resistor node N_(r)(10)(N−1)H 50(10).

In continuing reference to FIG. 15, and in both the exemplary and the alternative embodiments referenced above, the selected resistor node pair N_(r)(10)(i) 49(10) is selected based on the MSB code 48(10), the MSB code 48(10) has three (3) bits, N=2³=eight (8), R_(p)(10)=4R_(u), R_(frac)(10)=0.5R_(u), and R_(sd)(10)=36R_(u), R_(bulk1)(10)=R_(bulk2)(10)=0. Where MSB code 48(10)=i, the resistance between the selected resistor node pair N_(r)(10)(i) 49(10) is R_(p)(10)+R_(frac)(10) and meets the first interconnection relationship 68, R_(p)=R_(sd)∥(R_(p)+R_(frac)), as discussed in the previous figures. As the MSB code 48(10) is incremented, the resistance between the ground rail node V_(bot) 46(10) and the lower selected resistor node N_(r)(10)(i)L 52(10) should only increase by the primary resistance R_(p)(10) or 4R_(u). However, it will in fact increase by the primary resistance R_(p)(10)+the fractional resistance R_(frac)(10) or 4.5R_(u), and the third adjusting circuit 83(10) must incrementally remove the added fractional resistance from between the ground rail node V_(bot) 46(10) and the lower selected resistor node N_(r)(10)(i)L 52(10). One of the third fractional resistances R_(frac) 88(10)(i) is removed by closing one of the plurality of third switches 90(10)(i). In this manner, the resistance between the lower selected resistor node N_(r)(10)(i)L 52(10) and the ground rail node V_(bot) 46(10) will remain substantially constant and therefore meet the third interconnection relationship 74, i*R_(p)+R_(bulk1). This will have the effect of ensuring the voltage at the higher selected resistor node N_(r)(10)(N−1)H 50(10) of the last selection equals the lower selected resistor node N_(r)(10)(i)L 52(10) of the next selected resistor node pairs 49, N_(r)(10)(i)H 50(10) N_(r)(10)(i+1)L 52(10), as the MSB code 48(10) is incremented by one (1). As the MSB code 48(10) is decremented, the process will reverse itself and one of the plurality of third switches 90(10)(i) will open, therefore adding one of the plurality of third fractional resistances R_(frac) 88(10)(i) between the ground rail node V_(bot) 46(10) and the lower selected resistor node N_(r)(10)(i)L 52(10). In this manner, maintaining a constant and predictable voltage at the selected resistor node pair N_(r)(10)(i) 49(10) is possible.

As an example of the exemplary embodiment in FIG. 15, MSB code 48(10) has three (3) bits, N=2³=eight (8), R_(p)(10)=4R_(u), R_(frac)(10)=0.5R_(u), and R_(sd)(10)=36R_(u), R_(bulk1)(10)=R_(bulk2)(10)=0. When MSB code 48(10)=i=zero (0), the selected resistor node pair N_(r)(10)(0) 49(10) will be selected. The at least one current source 96 will maintain a constant current flow through the primary resistor string 40(10) in order to maintain constant divided voltages at each of the selectable resistor node pairs N_(r)(10)(0) to N_(r)(10)(N−1) 49(10). In the exemplary embodiment, the third adjusting circuit 83(10) will continue to maintain the resistance coupled between the ground rail node V_(bot) 46(10) and the lower selected resistor node N_(r)(10)(0)L 52(10) as discussed above. With MSB code 48(10)=zero (0), the third adjusting circuit 83(10) will initially begin with the plurality of third switches 90(10)(1) to 90(10)(N−1) open. In this manner, the plurality of the third fractional resistances R_(frac) 88(10)(1) to 88(10)(N−1) in the third adjusting circuit 83(10) will initially be included in the primary resistor string 40(10).

With continued reference to FIG. 15, as the MSB code 48(10) is incremented to one (1), the selected resistor node pair N_(r)(10)(1) 49(10) is then selected. An additional first fractional resistance R_(frac) 58(10) from the previous selected resistor node pair N_(r)(10)(0) 49(10) is now added into the primary resistor string 40(10) between the ground rail node V_(bot) 46(10) and the lower selected resistor node N_(r)(10)(1)L 52(10). The third adjusting circuit 83(10) compensates for this additional fractional resistance R_(frac) 58(10) by closing one of the plurality of third switches 90(10)(1). The closing of one of the plurality of third switches 90(10)(1) removes one of the plurality of third fractional resistances R_(frac) 88(10)(1) from the primary resistor string 40(10) between the ground rail node V_(bot) 46(10) and the lower selected resistor node N_(r)(10)(1)L 52(10). The exemplary embodiments in FIGS. 2-14 have a common goal to reduce the number and size of the required resistances and switches. The reduction in the number and size of the required resistances and switches is accomplished while still allowing the primary voltage divider 30 to be interconnected with the secondary voltage divider 32 without the isolation circuits VF1, VF2 found, as an example, in FIG. 1. One of the consequences of reducing the number and size of the required resistances and switches is the polarity of the coarse divided primary output voltage N_(r)(10)(i) 49(10) may reverse. The resulting reverse or flip in the voltage polarity of the primary voltage output would create trend reversals of the voltage output increase or decrease and the dual-string DAC 28 will not maintain monotonicity.

In this regard, embodiments disclosed herein also include polarity compensating dual-string DACs. Related circuits, systems, and methods are also disclosed. In embodiments disclosed herein, a secondary voltage divider of a dual-string DAC includes a switch logic unit. The switch logic unit is configured to compensate for polarity changes in the dual-string DAC to maintain monotonicity in the dual-string DAC. The dual-string DAC being montonic means that the dual-string DAC will convert a digital input code into an representative analog output voltage that increases (or stays constant) or decreases (or stays constant) for a given incremental change in the digital input code. Montonicity may be desired if it is desired for a DAC to convert digital codes to representative analog signals in a linear fashion. The incremental change in the DAC input code 15 may be either an increase or a decrease in the DAC input code 15 value. The switch logic unit is configured to select a secondary switch among the plurality of secondary switches to divide an input voltage from a primary voltage divider, based on a polarity indicator and a DAC input code. Each of the secondary switches is connected to a resistor node in a secondary resistor string of a secondary voltage divider. The switch logic unit is configured to select the secondary switch among the plurality of secondary switches to compensate for polarity changes in the input voltage from the primary voltage divider to the secondary resistor string. Thus, as a non-limiting example, the dual-string DAC can avoid the need to provide two switches for each resistor node in a primary resistor string to maintain monotonicity.

In this regard, the secondary voltage divider divides the V_(p) that has been selected and otherwise known as a selected primary DAC output voltage. The selected primary DAC output voltage is applied across the secondary resistor string and divided into finer secondary voltage levels. A finer secondary voltage level is selected and applied to the DAC output voltage V_(out) 38. The secondary switch unit comprises a plurality of secondary switches and the switch logic unit is comprised of a decoder and a polarity logic switching unit. In this manner, a polarity change in the voltage applied across the 2^(nd) rank or secondary resistor string may be compensated for, thus creating a dual-string DAC with a monotonic transfer function. The dual-string DAC maintains a monotonic transfer function even though the isolation circuits between the interconnected primary and secondary voltage dividers may have been eliminated. Eliminating isolation circuits will have the benefit of saving circuit size, semiconductor die area, cost, and performance increase. In the alternative, isolation circuits may not be eliminated.

For example, FIG. 16 illustrates a non-monotonic dual-string DAC 98 (referred to herein as “DAC 98”). A primary voltage divider 30(11) provides coarse divided primary voltage values by dividing the DAC input voltage (V_(dac) _(—) _(in)) across a plurality of primary resistors R(0)-R(N−1) in a primary resistor string 40(11) at selected resistor node pair N_(r) 49(11). A coarse divided primary voltage value is selected by a primary switch unit 42(11). The primary switch unit 42(11) selects a selected resistor node pair N_(r) 49(11) among the plurality of selected resistor node pairs N_(r)(0)-N_(r)(N) in the primary resistor string 40(11) as a selected coarse divided primary voltage V_(p). This selected coarse divided primary voltage V_(p) is then applied as V_(sec) _(—) _(in) across a secondary resistor string 60(11).

In this respect, with continuing reference to FIG. 16, the DAC 98 functions in a very similar fashion to the DAC 10 in FIG. 1. However, in order to properly convert the DAC input code 15 into the DAC output voltage V_(out) 38, the DAC 98 transfer function must remain monotonic throughout the range of possible digital codes. Remaining monotonic means the dual-string DAC is either monotonically increasing or monotonically decreasing. If the dual-string DAC is monotonically increasing, the voltage either increases or stays constant. If the dual-string DAC is monotonically decreasing, the voltage either decreases or stays constant. A monotonic transfer function means the selected coarse divided primary voltage V_(p) and the DAC output voltage V_(out) 38 either increases or stays constant for a monotonically increasing function or either decreases or stays constant for a monotonically decreasing function given an incremental change in a DAC input code 15. For example, as the DAC input code 15 increases in value, the selected coarse divided primary voltage V_(p) and the DAC output voltage V_(out) 38 must increase or remain constant (i.e., not decrease) for the DAC 98. Likewise, as the DAC input code 15 decreases in value, the selected coarse divided primary voltage V_(p) and the DAC output voltage V_(out) 38 must decrease or remain constant for the DAC 98. In the alternative, the selected coarse divided primary voltage V_(p) may increase or remain constant for an incrementally decreased DAC input code 15, where the DAC input code 15 has been inverted. Additionally, in the alternative, the selected coarse divided primary voltage V_(p) may decrease or remain constant for an incrementally increased DAC input code 15, where the DAC input code 15 has been inverted. The incremental adjustment to the DAC input code 15 may be either an increase or a decrease in the incremental adjustment. In both cases, a monotonic change in the selected coarse divided primary voltage V_(p) results. In FIG. 1, to provide monotonicity in the selected coarse divided primary voltage V_(p), primary switches U(0)-U(2N−1) are provided and configured to open and close to select resistor nodes N(0)-N_(r)(N) with no polarity changes or polarity flips. The primary switches U(0)-U(2N−1) are configured such that the voltage of the upper respective selected resistor node N_(r)(i)H 50 of the two (2) selected resistor node pair N_(r)(i) 49 is always greater. In FIG. 15, however, there is an issue when half of the primary switches U(0)-U(2N−1) are removed. The exemplary embodiment of FIG. 16 has only one (1) of the primary switches U(0)-U(2N−1) coupled to the selected resistor nodes N_(r)(0)-N_(r)(N). When the MSB code 48(11) is incremented or decremented by the primary switch unit 42(11) one selected resistor node is maintained and the other will skip to the next consecutive selected resistor node. This results in every other selection of selected resistor node pairs 49 alternating the voltage polarity of the selected coarse divided primary voltage V_(p).

As an example of the circuit in FIG. 16, the MSB code 48(11) code has a maximum value equal to N−1, where N=2^(MSB). In this example, the first selected resistor node pair 49 will be selected by MSB code 48(11)=N−1, or the maximum value. This selection causes the primary switch unit 42(11) to select the selected resistor node pair N_(r)(N) and N_(r)(N−1). The selected resistor node pairs N_(r)(N) and N_(r)(N−1) will be coupled to the selected coarse divided primary voltage V_(p) and then subsequently be further divided by the secondary voltage divider 32(11). The primary switch unit 42(11) causes the higher selected resistor node N_(r)(N) to be coupled to the coarse primary output voltage terminal 34O(11) and also cause the lower selected resistor node N_(r)(N−1)L 52(11) to be coupled to the coarse primary output voltage terminal 36O(11) terminal. However, a polarity reversal problem arises when the MSB code 48(11) is decremented to N−2. The polarity is then reversed, because the positive pole of V_(p) is now coupled to the selected resistor node N_(r)(N−1). The selected resistor node N_(r)(N−1) is coupled to the coarse primary output voltage terminal 36O(11) and the coarse primary output voltage terminal 34O(11) now coupled to selected resistor node N_(r)(N−2). Since the voltage at N_(r)(N−1) is greater than the voltage at N_(r)(N−2), the secondary DAC input voltage V_(sec) _(—) _(in) applied across the secondary DAC input voltage terminals 34I(11), 36I(11) is reversed and this pattern, flipping the positive and negative terminals, will continue as the MSB code 48(11) is further decremented. Because of the flipping or reversing of the polarity, a monotonic transfer function is lost.

In this regard, FIG. 17 is a generalized embodiment of a secondary voltage divider 32(12). The secondary voltage divider 32(12) is coupled to the selected coarse divided primary voltage V_(p) at a top node of the coarse primary output voltage terminal 34O(12), and a bottom node of the coarse primary output voltage terminal 36O(12). As discussed above, and as a non-limiting example, the coupling between the primary voltage divider (not shown) and the secondary voltage divider 32(12) is without any isolation circuits. The secondary voltage divider 32(12) comprises a secondary resistor string 60(12), a secondary switch unit 64(12), and a switch logic unit 100(12). The secondary resistor string 60(12) comprises a plurality of secondary resistors R_(s)(0)-R_(s)(Y−1) coupled in a serial configuration. In this manner, dividing a coarse primary output voltage V_(p) across the coarse primary output voltage terminals 34O(12), 36O(12) that has been selected is otherwise known as a selected primary DAC output voltage. The selected primary DAC output voltage V_(p) is applied across the secondary resistor string 60(12) into divided secondary voltage levels.

With continuing reference to FIG. 17, the secondary switch unit 64(12) comprises a plurality of secondary switches (not shown) that are each coupled to the secondary resistor string 60(12) at a plurality of secondary resistor nodes N_(sr)(0)-N_(sr)(N−1). The switch logic unit 100(12) is comprised of a decoder 102(12) and a polarity logic switching unit 104(12). The switch logic unit 100(12) is configured to receive the LSB code 66(12) and a polarity indicator 106. The MSB code 48 (not shown) and the LSB code 66(12) may be both incremented and decremented, causing a step up or a step down, selecting a correspondingly incremented or decremented secondary resistor node, as will be discussed below in FIG. 19. Each bit of the LSB code 66(12) corresponds to a voltage division step in the secondary resistor string 60(11). In one embodiment, the LSB code 66(12) is coupled to the decoder 102(12), and in an alternative embodiment the LSB code 66(12) is coupled to the polarity logic switching unit 104(12) as discussed below. In continuing reference to FIG. 16, the polarity indicator 106 is comprised of the least significant single bit of the MSB code 48.

The generalized embodiment of FIG. 17 operates by reversing the operation of the secondary switch unit 64(12) with every change in the setting of the polarity indicator 106. For example, when the polarity indicator 106 is zero (0), there would be no reversal of the secondary switches in the secondary switch unit 64(12). If there is no reversal of the secondary switches in the secondary switch unit 64(12), operation of the secondary switches (not shown) would incrementally select the next selected resistor node. The selection shifts from a bottom secondary resistor node N_(sr)(0) to a top secondary resistor node N_(sr)(N−1). However, if the polarity indicator 106 is set to one (1), this will indicate that the coarse primary output voltage across the coarse primary output voltage terminals 34O(12), 36O(12) has reversed polarity. In this manner, the operation of the secondary switch unit 64(12) will reverse. By reversing the selection, the next selected resistor node N_(sr)(i) will shift incrementally from the top resistor node N_(sr)(N−1) to the bottom resistor node N_(sr)(0). This coupling of the polarity indicator 106 and the LSB code 66(12) provides for a monotonic transfer function of the DAC input voltage (V_(dac) _(—) _(in)) and the DAC output voltage V_(out) 38.

In this regard, FIG. 18 illustrates a process of receiving the LSB code 66(12) and a polarity indicator 106 indicating a polarity of a secondary DAC input voltage V_(sec) _(—) _(in) applied across a secondary resistor string 60(12) (block 108). The secondary DAC input voltage V_(sec) _(—) _(in) applied across the secondary DAC input voltage terminals 34I(12), 36I(12) is the coarse primary output voltage across coarse primary output voltage terminals 34O(12), 36O(12). The selected coarse primary output voltage V_(p) at the coarse primary output voltage terminals 34O(12), 36O(12) is output from the primary voltage divider 30 and applied across the secondary resistor string 60(12) at the secondary DAC input voltage terminals 34I(12), 36I(12). Upon receiving the LSB code 66(12) and the polarity indicator 106, the switch logic unit 100 determines whether a reversal of the operation of the secondary switch unit 64(12) is necessary to maintain monotonicity. Based on the LSB code 66(12) and the polarity indicator 106, the secondary switch unit 64(12) selects a secondary switch within the secondary switch unit 64(12). The secondary switch unit 64 is coupled to a selected resistor node N_(sr)(0)-N_(sr)(N−1) within the secondary resistor string 60(12). The selection of the secondary switch within the secondary switch unit 64(12) causes the divided voltage at the selected resistor node N_(sr)(0)-N_(sr)(N−1) to apply the divided voltage to the DAC output voltage V_(out) 38 of the dual-string DAC 28 (block 110).

FIG. 19 is an exemplary embodiment of the process of FIG. 18. In FIG. 19, a secondary voltage divider 32(13) for a dual-string DAC 28 is provided that comprises a secondary resistor string 60(13), a secondary switch unit 64(13) and a switch logic unit 100(13). The secondary resistor string 60(13) comprises a plurality of secondary resistors R_(s)(0)-R_(s)(Y), where Y=2^(LSB). The secondary resistor string 60(13) further comprises a top secondary resistor R_(s)(Y) coupled to a top coarse primary output voltage terminal 34O(13), and a bottom secondary resistor R_(s)(0) coupled to a bottom coarse primary output voltage terminal 36O(13). The top secondary resistor R_(s)(Y) and the bottom secondary resistor R_(s)(0) are equal to half the value of each of the plurality of secondary resistors R_(s)(1)-R_(s)(Y−1). In the alternative, the top secondary resistor R_(s)(Y) and the bottom secondary resistor R_(s)(0) may be eliminated. The top secondary DAC input voltage terminal 34I(13) and the bottom secondary DAC input voltage terminal 36I(13) are configured to receive the selected coarse primary output voltage V_(p) applied as the secondary DAC input voltage V_(sec) _(—) _(in). The secondary DAC input voltage V_(sec) _(—) _(in) is applied across the secondary DAC input voltage terminals 34I(13), 36I(13). As discussed above, each bit of the LSB code 66(13) corresponds to a voltage division step in the secondary resistor string 60(13). It is therefore also true that the voltage division step across the secondary resistor R_(s)(0) and the secondary resistor R_(s)(Y) will be one-half (½) of each bit of the LSB code 66(13). The secondary switch unit 64(13) is comprised of a plurality of secondary switches 111 U(0) to U(Y−1), each coupled to a respective resistor node N_(sr)(0)-N_(sr)(N−1). The secondary switch unit 64(13) is further coupled to the switch logic unit 100(13). In this manner, the switch logic unit 100(13) controls the secondary switches 111 U(0)-U(Y−1). The switch logic unit 100(13) comprises a decoder 102(13) and a polarity logic switching unit 104(13). The decoder 102(13) is coupled to the plurality of secondary switches 111 U(0) to U(Y−1) and is also coupled to the polarity logic switching unit 104(13). The polarity logic switching unit 104(13) is coupled to the polarity indicator 106 and LSB code 66(13) and is also coupled to the decoder 102(13). The polarity logic switching unit 104(13) will control the reversal or non-reversal of the secondary switch unit 64(13), as will be explained below.

In this regard, to control the reversed or non-reversed operation of the switches, FIG. 19 also shows the polarity logic switching unit 104(13) comprising a plurality of Exclusive Or (XOR) logic gates 112. The XOR logic gates 112 comprise a first XOR input 114 and a second XOR input 116. The first XOR input 114 is coupled to the polarity indicator 106, and the second XOR input 116 is coupled to each one of the plurality of bits of the LSB code 66(13). The plurality of XOR logic gates 112 function to set each of a plurality of XOR logic gate outputs 118(13) to a one (1), if one and only one of the XOR logic gate inputs 114, 116 are set to one (1). The function of the polarity indicator 106 is to flip or reverse the corresponding bit of the plurality of corresponding bits output from the plurality of XOR logic gate outputs 118(13). The flip or reversal of the corresponding bit will occur when the polarity indicator 106 is set to one (1), indicating a reverse polarity mode. If a reverse polarity mode is set, the polarity logic switching unit 104(13) will reverse operation of the secondary switch unit 64(13). Reverse operation causes the resistor nodes N_(sr)(0) to N_(sr)(Y−1) to be selected sequentially in reverse order from the top secondary node N_(sr)(Y−1) to the bottom secondary node N_(sr)(0). Each of the plurality of XOR logic gate outputs 118(13) are coupled to a plurality of decoder inputs 120(13). The plurality of decoder inputs 120(13) may become a polarity modified LSB code 66(13) based on the polarity indicator 106. The combination of the plurality of decoder inputs 120(13) cause the decoder 102(13) to output a plurality of decoder outputs 122(13). The plurality of decoder outputs 122(13) control a secondary switch 111 U(0)-U(Y−1), as will be discussed in reference to the truth table in FIG. 20 below.

With continuing reference to FIG. 19, the secondary resistor string 60(13) comprises two (2) secondary resistors R_(s)(0) and R_(s)(Y) of a plurality of secondary resistors R_(s)(0)-R_(s)(Y). The two (2) secondary resistors R_(s)(0) and R_(s)(Y), as discussed above, are coupled to the top coarse primary output voltage terminal 34O(13) and the bottom coarse primary output voltage terminal 36O(13) respectively. The top secondary resistor R_(s)(Y) and the bottom secondary resistor R_(s)(0) are equal to half the value of each of the plurality of resistors R_(s)(1) to R_(s)(Y−1). The purpose for the modification of the resistor values equal to half of the remaining resistors R_(s)(1)-R_(s)(Y−1) is to compensate for the functional pivot of the secondary resistor string 60(13). The functional pivot of the secondary resistor string 60(13) occurs around the top secondary node N_(sr)(Y−1) or the bottom secondary node N_(sr)(0). The functional pivot of the secondary resistor string 60(13) occurs whenever the polarity indicator 106 indicates a reversal of the polarity.

In this regard, in FIG. 19, if the secondary resistors R_(s)(0) and R_(s)(Y) were valued at zero (0) and both the decoded MSB code 48 and the decoded LSB code 66 are equal to all ones (1111₂), the DAC output voltage V_(out) 38 is at the maximum. A problem would exist as the code counts down from <1111><0000>₂ to <1110><1111>₂, where both of these codes would select the voltage at the selected resistor node N_(r)(N−1) in FIG. 16. This would result in the voltage at secondary switches 111 U(0) and U(N−1) being equal. The problem of the two adjacent codes producing the DAC output voltage V_(out) 38, which is substantially equivalent, will happen at every carry or borrow between the MSB code 48 and LSB code 66. The problem of the two adjacent codes producing a DAC output voltage V_(out) 38 which is substantially equivalent will occur even though a step up or a step down in DAC output voltage V_(out) 38 is actually desired. The inclusion of the top secondary resistor R_(s)(Y) and bottom secondary resistor R_(s)(0), each equaling half the value of each of the plurality of resistors R_(s)(1) to R_(s)(Y−1), alleviates this problem. Each of these resistors will result with half LSB of voltage division regardless of the polarity of the voltage across the secondary voltage divider 32(13). Thus, when any of the MSB code 48 or LSB code 66 transitions, as indicated above, occur, there will be a total of 1 LSB of output voltage change. In this exemplary embodiment, the maximum output voltage of the DAC 28 would be the voltage at the top of the primary divider minus half LSB and the minimum output voltage would be half LSB above Vbot 36(13). In this manner, a monotonic and linear DAC transfer function is achieved.

The MSB code 48 and LSB code 66 transitions, the decoder inputs 120(13), the decoder outputs 122(13), and the resulting control of the secondary switches 111 U(0)-U(Y−1) of FIG. 19 may be illustrated as a truth table. The exemplary truth table in FIG. 20 illustrates an example of the secondary voltage divider 32(13) with a 4 bit LSB code 66. The exemplary truth table also shows how the polarity indicator 106 may cause the reverse polarity mode to reverse the operation of the secondary switch unit 64(13). If the polarity indicator 106 is set to zero (0), the non-reverse polarity mode is indicated and the LSB code 66 bits will not be changed. For instance, if the polarity indicator 106 is set to zero (0) and the corresponding LSB code 66 four (4) bits are 1011₂, according to the truth table, the plurality of XOR logic gate outputs 118 will be 1011₂. The XOR logic gate outputs 118 of 1011₂ would correspond to the closing of the secondary switch U(11). However, if the polarity indicator 106 indicates reverse polarity mode because it is set to one (1), then the plurality of XOR logic gate outputs 118 will be 0100₂, which would correspond to the closing of the secondary switch 111 U(4). The exemplary embodiment of FIG. 19 and the corresponding exemplary truth table of FIG. 20 illustrate the polarity switch logic driving the decoder 102, which in turn controls the secondary switch unit 64(13). In an alternative embodiment, it is possible to swap the function of the switch logic unit 100 and the decoder 102 such that the switch logic unit 100 controls the secondary switch unit 64(13) subsequent to the decoder 102 receiving the LSB code 66.

FIG. 21 shows an exemplary embodiment comprising a secondary resistor string 60(14), a secondary switch unit 64(14), and a switch logic unit 100(14). The secondary resistor string 60(14) comprises a plurality of secondary resistors R_(s)(0)-R_(s)(Y−1), where Y is equal to 2^(LSB) and LSB is the number of bits in the LSB code 66. The secondary resistor nodes N_(sr)(0)-N_(sr)(Y−2) are coupled to each of a plurality of secondary switches 111 U(0)-U(Y) in the secondary switch unit 64(14). The switch logic unit 100(14) comprises a decoder 102(14) configured to receive the LSB code 66 of the DAC input code 15, and decode the LSB code 66 to generate a DAC code selection output on one of a plurality of decoder outputs 122(14). The decoder 102(14) may be a LSB to a 2^(LSB) decoder. The switch logic unit 100(14) further comprises a plurality of multiplexers 123(14). Each of the plurality of multiplexers 123(14) comprises a first input 124(14), a second input 126(14), a control input 128(14), and a multiplexer output 130(14). The number of the plurality of multiplexers 123(14) may equal the number of the plurality of decoder outputs 122(14) plus one (1). The one (1) more than the number of the plurality of decoder outputs 122(14) accommodates a coupling to ground. The first input 124(14) is configured to receive a corresponding one of the plurality of decoder outputs 122(14) in non-reverse polarity mode. The second input 126(14) is configured to receive a corresponding one of the plurality of decoder outputs 122(14) in reverse mode. The first input 124(14) and the second input 126(14) are selected based on the control input 128(14) configured to receive the polarity indicator 106. In non-reverse polarity mode, each one of the decoder outputs 122(14), beginning with a first decoder output G0 corresponding to an LSB code 66 of 000₂, is coupled to a corresponding first input 124 of one of the plurality of multiplexers 123(14). The corresponding first input 124 of the plurality of multiplexers 123(14) in the three (3) bit example of LSB code 66 equaling 000₂ is Mux0. Each of the decoder outputs 122(14) from G1 to G(Y−1) is sequentially coupled to the first input 124(14) from Mux0 to Mux(Y). In reverse polarity mode, each one of the plurality of the decoder outputs 122(14), beginning with a last decoder output G(Y−1) corresponding to the LSB code 66 of 111₂ in a three (3) bit example, is coupled to a corresponding second input 126 of one of the plurality of multiplexers 123(14). The corresponding second input 126 of the plurality of multiplexers 123(14) in the three (3) bit example of LSB code 66 equaling 111₂ is Mux0. Each of the decoder outputs 122(14) from G(N−2) to G0 is sequentially coupled to the second input 126(14) from Mux0 to Mux(Y).

In continuing reference to FIG. 21, the decoder 102(14) will output on a corresponding one of the plurality of decoder outputs 122(14) based on the decoded result of the LSB code 66 input into the decoder 102(14). The one of the plurality of decoder outputs 122(14), G0 as an example, is coupled to the first input 124(14) of a first one of the plurality of multiplexers 123(14) and the second input 126(14) of a second one of the plurality of multiplexers 123(14). The polarity indicator 106 is coupled to the control input of each one of the plurality of multiplexers 123(14). The polarity indicator 106 can indicate a reverse polarity mode or a non-reverse polarity mode. If the polarity indicator 106 indicates a non-reverse polarity mode, the first one of the plurality of multiplexers 123(14) will pass through the switch selection to the corresponding secondary switch 111 U(0) to U(Y). If the polarity indicator 106 indicates reverse mode, the second one of the plurality of multiplexers 123(14) will pass through the switch selection to the corresponding secondary switch 111 U(0) to U(Y).

In this regard, FIG. 22 illustrates the exemplary truth table for FIG. 21 comprising values for the LSB code 66, the plurality of decoder outputs 122(14), the polarity indicator 106, and a corresponding secondary switch 111 U(0) to U(Y). The exemplary truth table illustrates the secondary voltage divider 32(14) with a three (3) bit LSB code 66. The exemplary truth table also illustrates how the polarity indicator 106 may cause the reverse polarity mode to reverse the operation of the secondary switch unit 64(14). If the polarity indicator 106 is set to zero (0), then non-reverse polarity mode is indicated and the LSB code 66 bits will not be changed. For example, if the polarity indicator 106 is set to zero (0) and the corresponding LSB code 66 is binary code of 101₂, the exemplary truth table indicates the output of G5 will be set of the plurality of decoder outputs 122(14). A decoder output 122 on G5 would correspond to the closing of secondary switch 111 U(5). However, if the polarity indicator 106 indicates reverse polarity mode with a setting of one (1) and the corresponding LSB code 66 is binary code 101, this would correspond to the closing of the secondary switch 111 U(2). The exemplary embodiment of FIG. 20 and the corresponding exemplary truth table of FIG. 21 illustrate the polarity switch logic driving the multiplexer 123(14), which in turn controls the secondary switch unit 64(13).

The dual-string DACs, and related circuits, systems, and methods according to embodiments disclosed herein may be provided in or integrated into any processor-based device. Examples, without limitation, include a set top box, an entertainment unit, a navigation device, a communications device, a fixed location data unit, a mobile location data unit, a mobile phone, a cellular phone, a computer, a portable computer, a desktop computer, a personal digital assistant (PDA), a monitor, a computer monitor, a television, a tuner, a radio, a satellite radio, a music player, a digital music player, a portable music player, a digital video player, a video player, a digital video disc (DVD) player, and a portable digital video player.

In this regard, FIG. 23 illustrates an example of a processor-based system 132 that can employ dual-string DACs 28 according to any of the embodiments disclosed herein. For example, the dual-string DACs 28 in the processor-based system 132 in FIG. 23 could include one or more adjusting circuits (not shown) configured to maintain the ideal voltage of the selected resistor node paired across the secondary voltage divider circuit 32 in the dual-string DAC 28. The dual-string DACs 28 in the processor-based system 132 in FIG. 23 could also include polarity compensating dual-string DACs 28 employing a switch logic unit configured to compensate for polarity changes in the dual-string DAC 28 to maintain monotonicity in the dual-string DAC 28. The dual-string DACs 28 in the processor-based system 132 in FIG. 22 could include both the aforementioned adjusting circuit(s) to maintain the ideal voltage of the selected resistor node paired across the secondary voltage divider circuit 32 in the dual-string DAC 28, and switch logic unit configured to compensate for polarity changes in the dual-string DAC 28 to maintain monotonicity in the dual-string DAC 28.

In this regard, the exemplary processor-based system 132 in FIG. 23 includes one or more central processing units (CPUs) 134 each including one or more processors 136. The CPU(s) 134, may have cache memory 138 coupled to the processor(s) 136 for rapid access to temporarily stored data. The CPU(s) 134 is coupled to a system bus 140 and can intercouple master and slave devices included in the processor-based system 132. As is well known, the CPU(s) 134 communicates with these other devices by exchanging address, control, and data information over the system bus 140. For example, the CPU(s) 134 can communicate bus transaction requests to a memory controller 142 as an example of a slave device. Although not illustrated in FIG. 23, multiple system buses 140 could be provided, wherein each system bus 140 constitutes a different fabric.

Other master and slave devices can be connected to the system bus 140. As illustrated in FIG. 23, these devices can include a memory system 144, one or more input devices 146, one or more output devices 148, one or more network interface devices 150, and one or more display controllers 152, as examples. The input device(s) 146 can include any type of input device, including but not limited to input keys, switches, voice processors, etc. The output device(s) 148 can include any type of output device, including but not limited to audio, video, other visual indicators, etc. The network interface device(s) 150 can be any devices configured to allow exchange of data to and from a network 154. The network 154 can be any type of network, including but not limited to a wired or wireless network, private or public network, a local area network (LAN), a wide local area network (WLAN), and the Internet. The network interface device(s) 150 can be configured to support any type of communication protocol desired. The memory system 144 can include one or more memory units 156(0-N). A bus interconnect arbiter 158 may be provided between the system bus 140 and master and slave devices coupled to the system bus 140, such as, for example, the memory units 156(0-N) provided in the memory system 144.

The CPU(s) 134 may also be configured to access the display controller(s) 152 over the system bus 140 to control information sent to one or more displays 160. The display controller(s) 152 sends information to the display(s) 160 to be displayed via one or more video processors 162, which process the information to be displayed into a format suitable for the display(s) 160. The display(s) 160 can include any type of display, including but not limited to a cathode ray tube (CRT), a liquid crystal display (LCD), a plasma display, etc.

Those of skill in the art will further appreciate that the various illustrative logical blocks, modules, circuits, and algorithms described in connection with the embodiments disclosed herein may be implemented as electronic hardware, instructions stored in memory or in another computer-readable medium and executed by a processor or other processing device, or combinations of both. The arbiters, master devices, and slave devices described herein may be employed in any circuit, hardware component, integrated circuit (IC), or IC chip, as examples. Memory disclosed herein may be any type and size of memory and may be configured to store any type of information desired. To clearly illustrate this interchangeability, various illustrative components, blocks, modules, circuits, and steps have been described above generally in terms of their functionality. How such functionality is implemented depends upon the particular application, design choices, and/or design constraints imposed on the overall system. Skilled artisans may implement the described functionality in varying ways for each particular application, but such implementation decisions should not be interpreted as causing a departure from the scope of the present disclosure.

The various illustrative logical blocks, modules, and circuits described in connection with the embodiments disclosed herein may be implemented or performed with a processor, a DSP, an Application Specific Integrated Circuit (ASIC), an FPGA or other programmable logic device, discrete gate or transistor logic, discrete hardware components, or any combination thereof designed to perform the functions described herein. A processor may be a microprocessor, but in the alternative, the processor may be any conventional processor, controller, microcontroller, or state machine. A processor may also be implemented as a combination of computing devices, e.g., a combination of a DSP and a microprocessor, a plurality of microprocessors, one or more microprocessors in conjunction with a DSP core, or any other such configuration.

The embodiments disclosed herein may be embodied in hardware and in instructions that are stored in hardware, and may reside, for example, in Random Access Memory (RAM), flash memory, Read Only Memory (ROM), Electrically Programmable ROM (EPROM), Electrically Erasable Programmable ROM (EEPROM), registers, a hard disk, a removable disk, a CD-ROM, or any other form of computer readable medium known in the art. An exemplary storage medium is coupled to the processor such that the processor can read information from, and write information to, the storage medium. In the alternative, the storage medium may be integral to the processor. The processor and the storage medium may reside in an ASIC. The ASIC may reside in a remote station. In the alternative, the processor and the storage medium may reside as discrete components in a remote station, base station, or server.

It is also noted that the operational steps described in any of the exemplary embodiments herein are described to provide examples and discussion. The operations described may be performed in numerous different sequences other than the illustrated sequences. Furthermore, operations described in a single operational step may actually be performed in a number of different steps. Additionally, one or more operational steps discussed in the exemplary embodiments may be combined. It is to be understood that the operational steps illustrated in the flow chart diagrams may be subject to numerous different modifications as will be readily apparent to one of skill in the art. Those of skill in the art will also understand that information and signals may be represented using any of a variety of different technologies and techniques. For example, data, instructions, commands, information, signals, bits, symbols, and chips that may be referenced throughout the above description may be represented by voltages, currents, electromagnetic waves, magnetic fields or particles, optical fields or particles, or any combination thereof.

The previous description of the disclosure is provided to enable any person skilled in the art to make or use the disclosure. Various modifications to the disclosure will be readily apparent to those skilled in the art, and the generic principles defined herein may be applied to other variations without departing from the spirit or scope of the disclosure. Thus, the disclosure is not intended to be limited to the examples and designs described herein, but are to be accorded the widest scope consistent with the principles and novel features disclosed herein. 

What is claimed is:
 1. A primary voltage divider of a dual-string digital-to-analog converter (DAC), comprising: a primary resistor string having a total resistance, the primary resistor string comprising a plurality of resistor nodes configured to divide a DAC input voltage applied across the primary resistor string into a plurality of divided voltage levels; a primary switch unit configured to receive a DAC input code and select a resistor node circuit among a plurality of resistor node circuits, the resistor node circuit comprising a selected resistor node pair among the plurality of resistor nodes of the primary resistor string based on the DAC input code to couple a divided voltage level across the selected resistor node pair to a secondary voltage divider circuit of a dual-string DAC; and at least one adjusting circuit comprising at least one first fractional resistance configured to adjust a resistance of the selected resistor node pair to maintain an ideal voltage of the selected resistor node pair across the secondary voltage divider circuit in response to the primary switch unit selecting the selected resistor node pair, without impedance isolation between the primary resistor string and the secondary voltage divider circuit.
 2. The primary voltage divider of claim 1 not comprising an isolating circuit coupled between the primary voltage divider and the secondary voltage divider circuit.
 3. The primary voltage divider of claim 1, wherein the at least one adjusting circuit is comprised of at least one first adjusting circuit providing the at least one first fractional resistance to the selected resistor node circuit, the selected resistor node circuit comprising the at least one first adjusting circuit.
 4. The primary voltage divider of claim 3, wherein: a resistance of the selected resistor node pair is comprised of a primary resistance (R_(p)) and the at least one first fractional resistance (R_(frac1)); a resistance of the secondary voltage divider circuit is comprised of resistances providing a total secondary voltage divider circuit resistance (R_(sd)); and the primary resistance (R_(p)) is 1/(1/(R_(p)+R_(frac1))+(1/R_(sd))).
 5. The primary voltage divider of claim 3, wherein the at least one adjusting circuit further comprises: a second adjusting circuit configured to include or not include at least one second fractional resistance in the total resistance of the primary resistor string in response to the primary switch unit selecting a resistor node pair; and a third adjusting circuit configured to include or not include at least one third fractional resistance in the total resistance of the primary resistor string in response to the primary switch unit selecting the resistor node pair; and the second adjusting circuit and the third adjusting circuit configured to maintain the ideal voltage of the selected resistor node pair across the secondary voltage divider circuit, in response to the primary switch unit selecting the resistor node pair.
 6. The primary voltage divider of claim 5, wherein: the second adjusting circuit further comprises at least one second switch configured to include or not include the at least one second fractional resistance in the total resistance of the primary resistor string; and the third adjusting circuit further comprises at least one third switch configured to include or not include the at least one third fractional resistance in the total resistance of the primary resistor string.
 7. The primary voltage divider of claim 5, wherein: the second adjusting circuit is further configured to adjust a resistance of the primary resistor string between a voltage rail node of the primary resistor string and the selected resistor node circuit to adjust the total resistance of the primary resistor string; and the third adjusting circuit is further configured to adjust a resistance of the primary resistor string between a ground rail node of the primary resistor string and the selected resistor node circuit to adjust the total resistance of the primary resistor string.
 8. The primary voltage divider of claim 7, wherein: the resistance of the selected resistor node pair is comprised of the primary resistance (R_(p)) and the at least one first fractional resistance (R_(frac1)); the DAC input code (i) is comprised of a plurality of binary input bits (n), each combination of the plurality of binary input bits configured to select a unique resistor node pair in the primary resistor string; the resistance of the primary resistor string between the voltage rail node of the primary resistor string and the selected resistor node circuit is represented by (N-i-1)*R_(p); and the resistance of the primary resistor string between the ground rail node of the primary resistor string and the selected resistor node circuit is represented by i*R_(p).
 9. The primary voltage divider of claim 8, further comprising: a second biasing resistance (Rbulk2) configured to vary the total resistance of the second adjusting circuit; and a third biasing resistance (Rbulk1) configured to vary the total resistance of the third adjusting circuit; wherein: the resistance of the primary resistor string between the voltage rail node of the primary resistor string and the selected resistor node circuit is represented by (N-i-1)*R_(p)+R_(bulk2); and the resistance of the primary resistor string between the ground rail node of the primary resistor string and the selected resistor node circuit is represented by i*R_(p)+R_(bulk1).
 10. The primary voltage divider of claim 3, wherein the selected resistor node pair comprises a primary resistance coupled in series with the at least one first adjusting circuit.
 11. The primary voltage divider of claim 3, wherein the at least one first adjusting circuit further comprises at least one first switch configured to couple the at least one first fractional resistance to the selected resistor node pair when the at least one first switch is switched in a coupling mode.
 12. The primary voltage divider of claim 3, wherein the at least one first adjusting circuit further comprises at least one first switch configured to not couple the at least one first fractional resistance to the selected resistor node pair when the at least one first switch is switched in a decoupling mode.
 13. The primary voltage divider of claim 3, wherein: the at least one first fractional resistance is comprised of a shared fractional resistance; and the at least one adjusting circuit is comprised of at least one first adjusting circuit, wherein the at least one first adjusting circuit is comprised of at least one first switch selectively controlled by the DAC input code to provide the shared fractional resistance to the selected resistor node circuit.
 14. The primary voltage divider of claim 5, wherein: the second adjusting circuit further comprises at least one second switch configured to include or not include the at least one second fractional resistance in the total resistance of the primary resistor string; the second adjusting circuit further configured to adjust a resistance of the primary resistor string between a voltage rail node of the primary resistor string and the selected resistor node circuit; the third adjusting circuit further comprises at least one third switch configured to include or not the at least one third fractional resistance in the total resistance of the primary resistor string; and the third adjusting circuit further configured to adjust a resistance of the primary resistor string between a ground rail node of the primary resistor string and the selected resistor node circuit.
 15. The primary voltage divider of claim 14, wherein: the at least one second fractional resistance of the second adjusting circuit comprises a plurality of second fractional resistances coupled to each other in series; the at least one second switch of the second adjusting circuit is further configured to include a second fractional resistance of the plurality of second fractional resistances in the total resistance of the primary resistor string in response to an increase in the DAC input code; the at least one third fractional resistance of the third adjusting circuit comprising a plurality of third fractional resistances coupled to each other in series; and the at least one third switch of the third adjusting circuit is further configured to not include a third fractional resistance of the plurality of third fractional resistances in the total resistance of the primary resistor string for each increase in the DAC input code.
 16. The primary voltage divider of claim 15, wherein: the at least one second switch of the second adjusting circuit is further configured to not include the second fractional resistance of the plurality of second fractional resistances in the total resistance of the primary resistor string in response to a decrease in the DAC input code; and the at least one third switch of the third adjusting circuit is further configured to include the third fractional resistance of the plurality of third fractional resistances in the total resistance of the primary resistor string for each decrease in the DAC input code.
 17. The primary voltage divider of claim 14, wherein the at least one first fractional resistance is common among at least two of the plurality of resistor node circuits.
 18. The primary voltage divider of claim 5, wherein: the second adjusting circuit further comprises a single second switch configured to include or not include a single second fractional resistance in the total resistance of the primary resistor string between a voltage rail node of the primary resistor string and the selected resistor node circuit; and the third adjusting circuit further comprises a single third switch configured to include or not include a single third fractional resistance in the total resistance of the primary resistor string between a ground rail node of the primary resistor string and the selected resistor node circuit.
 19. The primary voltage divider of claim 18, wherein the at least one first fractional resistance of the selected resistor node circuit is common among at least two of the plurality of resistor node circuits.
 20. The primary voltage divider of claim 3, wherein the at least one adjusting circuit comprises at least one current source coupled to the primary voltage divider configured to maintain the ideal voltage of the selected resistor node pair across the secondary voltage divider circuit.
 21. The primary voltage divider of claim 20, wherein the at least one adjusting circuit comprises: a second adjusting circuit configured to selectively include at least one second fractional resistance in the total resistance of the primary resistor string in response to the primary switch unit selecting the resistor node pair; and the second adjusting circuit configured to maintain the ideal voltage of the selected resistor node pair across the secondary voltage divider circuit in response to the primary switch unit selecting the resistor node pair.
 22. The primary voltage divider of claim 21, wherein: a resistance of the selected resistor node pair is comprised of a primary resistance (R_(p)) and a first fractional resistance (R_(frac1)); a resistance of the secondary voltage divider circuit is comprised of resistances providing a total secondary voltage divider circuit resistance (R_(sd)); the primary resistance (R_(p)) is 1/(1/(R_(p)+R_(frac1))+(1/R_(sd))); and the at least one current source adjusts a current (I) so that the ideal voltage (V_(ideal)) is equal to an actual voltage (V_(actual)), where V_(actual)=I*1/(1(R_(p)+R_(frac1))+(1/R_(sd))).
 23. The primary voltage divider of claim 1, wherein the at least one adjusting circuit comprises at least one current source coupled to the secondary voltage divider circuit configured to maintain the ideal voltage of the selected resistor node pair across the secondary voltage divider circuit.
 24. The primary voltage divider of claim 23, wherein: a resistance of the selected resistor node pair is comprised of a primary resistance (R_(p)); a resistance of the secondary voltage divider circuit is comprised of resistances providing a total secondary voltage divider circuit resistance (R_(sd)); the primary resistance (R_(p)) is 1/(1/(R_(p)+R_(frac1))+((1/R_(sd))); and the at least one current source adjusts a current (I) so that the ideal voltage (V_(ideal)) is equal to an actual voltage (V_(actual)), where V_(actual)=I*1/(1/(R_(p)+R_(frac1))+((1/R_(sd))).
 25. The primary voltage divider of claim 1 integrated into a semiconductor die.
 26. The primary voltage divider of claim 1, further comprising a device selected from the group consisting of a set top box, an entertainment unit, a navigation device, a communications device, a fixed location data unit, a mobile location data unit, a mobile phone, a cellular phone, a computer, a portable computer, a desktop computer, a personal digital assistant (PDA), a monitor, a computer monitor, a television, a tuner, a radio, a satellite radio, a music player, a digital music player, a portable music player, a digital video player, a video player, a digital video disc (DVD) player, and a portable digital video player, into which the primary voltage divider is integrated.
 27. A primary voltage divider of a dual-string digital-to-analog converter (DAC) for dividing a total voltage across a series of resistive nodes, wherein the primary voltage divider comprises: a means for dividing a total voltage, a primary resistor string having a total resistance, the primary resistor string comprising a plurality of resistor nodes configured to divide a DAC input voltage applied across the primary resistor string into a plurality of divided voltage levels; a means for selecting a resistor node circuit comprising a selected resistor node pair among the plurality of resistor nodes of the primary resistor string based on a DAC input code to couple a divided voltage level across the selected resistor node pair to a secondary voltage divider circuit of a dual-string DAC; and a means for adjusting a resistance of the selected resistor node pair with at least one first fractional resistance to maintain an ideal voltage of the selected resistor node pair across the secondary voltage divider circuit in response to a primary switch unit selecting the selected resistor node pair.
 28. A method of dividing voltage in a dual-string digital-to-analog converter (DAC) comprising: dividing a total voltage, a primary resistor string having a total resistance, the primary resistor string comprising a plurality of resistor nodes configured to divide a DAC input voltage applied across the primary resistor string into a plurality of divided voltage levels; selecting a resistor node circuit comprising a selected resistor node pair among the plurality of resistor nodes of the primary resistor string based on a DAC input code to couple a divided voltage level across the selected resistor node pair to a secondary voltage divider circuit of a dual-string DAC; and adjusting a resistance of the selected resistor node pair with at least one first fractional resistance to maintain an ideal voltage of the selected resistor node pair across the secondary voltage divider circuit in response to a primary switch unit selecting the selected resistor node pair. 